This paper constructs a convex set representing the moduli space of certain geometric manifolds and characterizes regular boundary behavior, with applications to deformations in supergravity theories.
Contribution
It introduces a compact convex set for the moduli of projective special real manifolds and characterizes their boundary behavior, advancing understanding of their deformation theory.
Findings
01
Constructed a convex generating set $\
02
$ of the moduli space.
03
Characterized when a manifold has regular boundary behavior.
Abstract
We construct a compact convex generating set Cn of the moduli set of closed connected projective special real manifolds of fixed dimension n. We show that a closed connected projective special real manifold corresponds to an inner point of Cn if and only if it has regular boundary behaviour. Our results can be used to describe deformations of 5d supergravity theories with complete scalar geometries.
Equations577
Cn:={x3−x⟨y,y⟩+P3(y)∥y∥=1maxP3(y)≤332}
Cn:={x3−x⟨y,y⟩+P3(y)∥y∥=1maxP3(y)≤332}
{x3−x⟨y,y⟩+P3(y)P3∈Sym3(Rn)∗}⊂Sym3(Rn+1)∗.
{x3−x⟨y,y⟩+P3(y)P3∈Sym3(Rn)∗}⊂Sym3(Rn+1)∗.
h:H1,1(X,R)→R,[ω]↦∫Xω3.
h:H1,1(X,R)→R,[ω]↦∫Xω3.
df(TpM)⊕Rξf(p)=Tf(p)Rn+1
df(TpM)⊕Rξf(p)=Tf(p)Rn+1
∇X(df(Y))=df(∇XY)+g(X,Y)ξf,
∇X(df(Y))=df(∇XY)+g(X,Y)ξf,
−∂2hp(⋅,⋅)=−∂2hAp(A⋅,A⋅)=A∗(−∂2h)p.
−∂2hp(⋅,⋅)=−∂2hAp(A⋅,A⋅)=A∗(−∂2h)p.
gH=−τ1∂2h∣TH×TH,
gH=−τ1∂2h∣TH×TH,
{[H]∣H is a CCGPSR manifold of degree τ,dim(H)=n},
{[H]∣H is a CCGPSR manifold of degree τ,dim(H)=n},
U=R>0⋅H={rp∈Rn+1r>0,p∈H}⊂Rn+1
U=R>0⋅H={rp∈Rn+1r>0,p∈H}⊂Rn+1
(p+TpH)∩U⊂p+TpH
(p+TpH)∩U⊂p+TpH
h=xτ+xτ−1L(y)+xτ−2Q(y,y)+(terms of lower order in x),
h=xτ+xτ−1L(y)+xτ−2Q(y,y)+(terms of lower order in x),
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TopicsBlack Holes and Theoretical Physics · Noncommutative and Quantum Gravity Theories · Homotopy and Cohomology in Algebraic Topology
Full text
**Properties of the moduli set of complete
connected projective special real manifolds
**
David Lindemann
Department of Mathematics and Center for Mathematical Physics
University of Hamburg,
Bundesstraße 55, D-20146 Hamburg, Germany
We construct a compact convex generating set Cn of the moduli set of closed connected projective special real manifolds of fixed dimension n. We show that a closed connected projective special real manifold corresponds to an inner point of Cn if and only if it has regular boundary behaviour. Our results can be used to describe deformations of 5d supergravity theories with complete scalar geometries.
**Keywords: affine differential geometry, centro-affine hypersurfaces, Kähler cones, projective special real manifolds, special geometry
Projective special real (short: PSR) manifolds are centro-affine hypersurfaces that are contained in a positive level set of a hyperbolic homogeneous cubic polynomial and consist only of hyperbolic points of said polynomial. Their study is closely related to the theory of supergravity in five spacetime dimensions, where they are the scalar manifolds [GST, DV]. By the so-called supergravity r-map and c-map their geometry is related to the study of projective special Kähler manifolds [ACD, F] and quaternionic Kähler manifolds [DV].
Completeness properties of the r- and c-map have been studied in [CHM] where it was shown that both the supergravity r-map (which associates to an n-dimensional PSR manifold a 2n+2-dimensional projective special Kähler manifold) and the supergravity c-map (which associates to an m-dimensional projective special Kähler domain a 2m+4-dimensional quaternionic Kähler manifold) preserve geodesic completeness. This yields an explicit way to obtain examples of complete non-compact quaternionic Kähler manifolds of negative scalar curvature by taking a complete connected PSR manifold and composing the supergravity r- and c-map (this is called the supergravity q-map [DV]). Note that it has been proven in [FS] that all manifold in the image of the supergravity c-map have negative scalar curvature, independent of completeness. In [CNS], hyperbolic centro-affine hypersurfaces have been studied and it was shown PSR manifolds are geodesically complete if and only if they are closed in their ambient space [CNS, Thm. 2.5].
The mentioned results motivate the study of the moduli set of all closed connected PSR manifolds of fixed dimension n. By the term “moduli set” we mean the set of isomorphism classes of closed PSR manifolds. Two closed connected PSR manifold are isomorphic if they are contained in the same GL(n+1)-orbit, where the corresponding GL(n+1)-action is induced by the action on the ambient space Rn+1. We will call two isomorphic PSR manifolds equivalent. In dimension 1 and 2, all PSR manifolds without the restriction of being closed in the ambient space have been completely classified up to equivalence, cf. [CHM] for curves and [CDL] for surfaces, but in higher dimensions no complete classification has been found yet. However, there exist classification results when further restricting the geometry. PSR manifolds that are homogeneous spaces have been classified in [DV], and PSR manifolds related to reducible cubic polynomials have been classified in [CDJL].
Our main result in this work is the construction of a generating set of the moduli set of closed connected PSR manifolds in all dimensions:
Theorem 1.1**.**
Let (xy)=(x,y1,…,yn)T denote linear coordinates on Rn+1, let ⟨⋅,⋅⟩ denote the standard Euclidean scalar product on Rn induced by the choice of the coordinates (y1,…,yn)T, and let ∥⋅∥ denote the corresponding norm.
For all n∈N, the set of hyperbolic homogeneous cubic polynomials
[TABLE]
is a generating set for the moduli set of n-dimensional closed connected PSR manifolds under the action of GL(n+1), meaning that for every closed connected PSR manifold H of dimension n there exists an element h∈Cn, such that the connected component H⊂{h=1} containing the point (xy)=(10)∈{h=1}⊂Rn+1 is equivalent to H
and, conversely, each h∈Cn defines a closed connected PSR manifold which is given by the connected component of {h=1} that contains the point (xy)=(10).
The set Cn⊂Sym3(Rn+1)∗ is a uniformly bounded compact convex subset of the 6n3+3n2+2n-dimensional affine subspace
[TABLE]
The boundary of Cn, that is ∂Cn, is a continuous submanifold of Sym3(Rn+1)∗.
Furthermore, h∈∂Cn if and only if the initial H does not have regular boundary behaviour.
As we will describe in more detail in Remark 4.16, it is in general not difficult to construct a bounded generating set for the moduli set of closed connected PSR manifolds. However, it was up until now not clear that we can find a bounded generating set of dimension less than the dimension of the vector space of cubic homogeneous polynomials in n+1 variables (corresponding to n-dimensional PSR manifolds), and the existence of such a generating set that is additionally closed and convex is also far from obvious. Furthermore, points in ∂Cn correspond precisely to closed connected PSR manifolds that do not have regular boundary behaviour. A PSR manifold H⊂{h=1}⊂Rn+1 having regular boundary behaviour means that
the negative Hessian of h, −∂2h, viewed as a bilinear form on Rn+1 has only 1-dimensional kernel along ∂U∖{0}, and dhp=0 for all p∈∂U∖{0}, where U:=R>0⋅H⊂Rn+1 denotes the cone spanned by H.
The main ingredient in order to show that points in ∂Cn correspond precisely to closed connected PSR manifolds with non-regular boundary behaviour is Theorem 4.12, where we prove that for a closed connected PSR manifold H⊂{h=1} it suffices to show that dhp=0 for all p∈∂U∖{0} in order to show that H has regular boundary behaviour.
Note at this point that we consider the moduli set of PSR manifolds as a set, not as a topological space and, hence, we do not use the term “moduli space”. Choosing and describing a suitable topology on the moduli set of PSR manifolds, which would then justify the term moduli space, is an interesting task for future studies.
While the generating set Cn is not a 1:1 description of the moduli set, its compactness and convexity properties imply many new properties of the moduli set. Compactness shows that for fixed dimension, the sectional curvatures and scalar curvature of complete PSR manifolds are bounded from above and below with bounds depending only on the dimension, see Corollary 5.1. To prove said result we will develop curvature formulas for PSR manifolds (cf. Proposition 3.9 for a larger class of manifolds outlined below) and use a technical result for a standard form of PSR manifolds, see Proposition 3.1. Convexity enables us to explicitly describe a curve in the class of closed connected PSR manifolds of dimension n that connects any two given closed connected PSR manifolds of dimension n, see Corollary 4.17 and Example 4.18. The compactness property of Cn can also be used to find an alternative proof that closed PSR manifolds are complete, see Proposition 5.3. Furthermore, these properties also carry over to the supergravity r- and q-map and thereby in particular yield an explicit way to deform two quaternionic Kähler manifolds in the image of the supergravity q-map (restricted to closed connected PSR manifolds) into each other, which is by Theorem 1.1 always possible independent of their initial choice. This in particular means that we have developed a way to deform two distinct 5d supergravity theories, which are complete in the sense that their scalar manifolds (which are PSR manifolds) are geodesically complete, into each other via a curve of such theories which corresponds point-wise to other (but possibly equivalent) theories that are also complete in the stated sense. This property also carries over to 4d and 3d supergravity theories obtainable via the supergravity r- and q-map, respectively.
In this work we also study what we call generalized PSR manifolds (GPSR manifolds for short). A GPSR manifold (of degree τ) is a centro-affine hypersurface contained in the level set of a hyperbolic homogeneous polynomial of degree τ≥3, so in comparison with PSR manifolds (which we view as a subclass of GPSR manifolds) we also allow polynomial of degree higher than 3. GPSR manifolds with corresponding polynomial of degree τ≥4 do not have a similar motivation from supergravity, but as for PSR manifolds they appear as level sets in the Kähler cones (or, more generally, index/positive cones) of compact Kähler τ-folds. Many of our technical results like Proposition 3.1 and Proposition 3.12 hold for GPSR manifolds of arbitrary degree τ≥3. However, it turned out to be very hard to prove a statement similar to Theorem 1.1 for GPSR manifolds of degree τ≥4. In [Li, Thm. 7.2] closed connected GPSR curves of degree 4 have been classified up to isomorphisms, and it turned out that not only was this far more complicated than an analogous classification of closed connected PSR curves (cf. [Li, Rem. 7.4], which proceeds differently than the complete classification of PSR curves in [CHM]), but also that even in this most simple case of GPSR manifolds of degree ≥4 we cannot find a compact convex generating set in the way we did for closed connected PSR manifolds. In future studies, we will expand the work on GPSR manifolds of degree 4 that was done in [Li, Sect. 7]. In particular, it is an open problem whether or not such manifolds are geodesically complete if they are closed in the ambient space. The latter property has first been proven for closed PSR manifolds in [CNS, Thm. 2.5] and we will present an alternative proof in Proposition 5.3 which uses our main Theorem 1.1. Another interesting open question is the generalization of the supergravity q-map to connected GPSR manifolds of degree τ≥4.
Aside from the study of supergravity, PSR manifolds also appear in the study of compact Kähler threefolds. For a compact Kähler threefold X consider the cubic homogeneous polynomial
[TABLE]
By the Hodge-Riemann bilinear relations, we find that each point in the cone of Kähler classes K⊂H1,1(X,R) is a hyperbolic point of h, and so the set H:={h=1}∩K is a PSR manifold. This point of view has been studied in particular for Calabi-Yau threefolds. In [W, TW], the sectional curvatures of, albeit not under this name, PSR manifolds have been studied from this point of view and in [KW] the defining trilinear forms and their relation to the second and third Chern classes have been studied.
Analogously, GPSR manifolds of degree τ≥4 appear as level sets in cones of Kähler classes of compact Kähler manifolds X of complex dimension τ. The corresponding hyperbolic homogeneous polynomial is then given by h:[ω]↦∫Xωτ. For τ=4, the curvature of a GPSR manifold (again, not under this name) has been studied in [T, Sect. 4]. An expanded study of the different curvatures of GPSR manifolds, with applications to both the geometry of Kähler cones and supergravity, will be the subject of an upcoming paper.
Acknowledgements
This work was partly supported by the German Science Foundation (DFG) under the Research Training Group 1670 and the Collaborative Research Center (SFB) 676.
It is based on the main results of Section 5 and technical results of Sections 3 and 4 of my doctoral thesis. Proposition 4.6 is new and not part of my doctoral thesis. I would like to thank my supervisor Vicente Cortés for his continuous support during the writing of my thesis, and I would also like to thank my second examiners Andriy Haydys and Antonio Martínez.
2 Preliminaries
We will quickly review the basics of centro-affine differential geometry that are needed in this work.
Definition 2.1**.**
Let f=(f1,…,fn+1)T:M→Rn+1 be a hypersurface immersion. It is called a centro-affine hypersurface immersion if the position vector field ξ∈Γ(TRn+1), ξp=p for all p∈Rn+1 under the canonical identification, is transversal along f, that is
[TABLE]
for all p∈M, where Rξf(p) denotes the 1-dimensional vector subspace spanned by ξf(p) of Tf(p)Rn+1. We will omit writing down the map f if it is clear from the context and simply call M a centro-affine hypersurface.
If f is additionally an embedding, it will be called a centro-affine hypersurface embedding.
The Gauß equation for centro-affine hypersurface immersions f:M→Rn+1 is of the form
[TABLE]
where ξf denotes the position vector field along f. This leads to the following definition.
Definition 2.2**.**
Let f:M→Rn+1 be a centro-affine hypersuface immersion. The induced connection ∇ in TM (2.1) is called the centro-affine connection, the symmetric (0,2)-tensor g∈Γ(Sym2T∗M) is called the centro-affine fundamental form. The centro-affine hypersurface immersion f is called non-degenerate if g is non-degenerate, definite if g is definite, i.e. either positive or negative definite, elliptic if g<0, i.e. negative definite, and hyperbolic if g>0, i.e. positive definite.
A tool to obtain explicit examples of centro-affine hypersurfaces together with their centro-affine fundamental form is explained in the following proposition.
Proposition 2.3**.**
Let U⊂Rn+1, n∈N∪{0}, be open and invariant under positive rescaling, i.e. rp∈U for all r>0 and p∈U. Let h:U→R be a homogeneous function of degree k>1. Assume that the level set {p∈U∣h(p)=1} is not empty and let H⊂{p∈U∣h(p)=1} be open subset. Then the inclusion map ι:H→Rn+1 is a centro-affine hypersurface embedding with centro-affine fundamental form g=−k1ι∗(∇2h), where ∇ denotes the canonical flat connection in TRn+1.
Proof.
For a proof of this statement in a slightly more general setting see [CNS, Prop. 1.3].
∎
If Rn+1 is equipped with linear coordinates, we will write ∂2 instead of ∇2. We will also omit writing down the map ι for an embedding ι:M→Rn+1, that is we will write M⊂Rn+1 instead of ι(M)⊂Rn+1. In this work we will study centro-affine hypersurface embeddings where h as in Proposition 2.3 is a hyperbolic homogeneous polynomial of degree τ≥3 with Riemannian centro-affine fundamental form.
Definition 2.4**.**
Let U⊂Rn+1 be an open subset that is invariant under multiplication with positive real numbers, and let h:U→R be a homogeneous function of degree τ>1. Then a point p∈{h>0} is called a hyperbolic point (of h) if −∂2hp has signature (n,1), i.e. it is of Lorentz type. A function h that has at least one hyperbolic point is called a hyperbolic homogeneous function.
Observe that this implies that for a hyperbolic point p of h we have −∂2hp∣ker(dhp)×ker(dhp)>0, which follows from −∂2hp(p,p)=−τ(τ−1)h(p)<0 and −∂hp(p,⋅)=−(τ−1)dhp.
Definition 2.5**.**
Let H⊂{h=1} be a centro-affine hypersurface as in Proposition 2.3. Then H is called a hyperbolic centro-affine hypersurface if it consists only of hyperbolic points.
Note that the above definition of hyperbolic centro-affine hypersurface coincides with Definition 2.2 for f the inclusion map ι:H→Rn+1. Hyperbolic centro-affine hypersurfaces equipped with their respective centro-affine fundamental form (H,g) are Riemannian manifolds. Continuity of the determinant implies that a connected non-degenerate centro-affine hypersurface H is hyperbolic if and only if it contains one hyperbolic point. Note that hyperbolicity at a point is an open condition in the sense every homogeneous function h:U→R as in Definition 2.4 with a hyperbolic point p is hyperbolic on some open neighbourhood V⊂U of p, which also follows from the continuity of the determinant of −∂2h. Hence, for every hyperbolic homogeneous function h of degree τ>1 we can choose an open subset H⊂{h=1} that is a hyperbolic centro-affine hypersurface.
Definition 2.6**.**
A homogeneous polynomial h:Rn+1→R of degree τ≥2 is called a hyperbolic homogeneous polynomial if there exists p∈{h>0}, such that p is a hyperbolic point of h.
Next, we will introduce a notion of equivalence for homogeneous polynomials and hypersurfaces in their respective level sets.
Definition 2.7**.**
Two hyperbolic homogeneous polynomials h,h:Rn+1→R of degree τ≥2 are called equivalent if there exists a linear transformation A∈GL(n+1), such that h∘A=h. Two connected hyperbolic centro-affine hypersurfaces H and H contained in a level set of h and h, respectively, are called equivalent if h and h are equivalent and A(H)=H for h∘A=h.
Two centro-affine hyperbolic hypersurfaces being equivalent in particular implies the following statement.
Lemma 2.8**.**
Any two equivalent connected centro-affine hyperbolic hypersurfaces H and H defined by hyperbolic homogeneous polynomials of degree τ≥2h,h:Rn+1→R as in Definition 2.7, respectively, are isometric.
Proof.
Let A∈GL(n+1), such that h∘A=h. Since A:Rn+1→Rn+1 is linear, we get
[TABLE]
By restricting the above equation to TH, respectively TH, we find with Proposition 2.3 that H and H are indeed isometric and one isometry is given by the linear transformation A relating the two polynomials h and h.
∎
An additional topological property of centro-affine hypersurfaces that we will often impose is the following.
Definition 2.9**.**
A connected hyperbolic centro-affine hypersurface H⊂{h=1} is called maximal if it is a connected component of the set {h=1}∩{hyperbolic points of h}.
Now we will introduce the centro-affine hypersurfaces that are our main focus of study in this work.
Definition 2.10**.**
Let n∈N∪{0} and H⊂{h=1}⊂Rn+1 be an n-dimensional hyperbolic centro-affine hypersurface as in Definition 2.5 contained in the level set of a hyperbolic homogeneous polynomial of degree τ≥3. Then H will be called a GPSR manifold (for Generalised Projective Special Real manifold) of degree τ. If we further assume that H is closed and connected as a subset of Rn+1, we will call H a CCGPSR manifold (for Closed Connected GPSR manifold) of degree τ.
For τ=3, we will call H a PSR manifold, or a CCPSR manifold if it is closed in its ambient space.
As a convention we regard the set of CCPSR manifolds as a subset of the set of CCGPSR manifolds.
If the degree τ≥3 of a GPSR manifold is not of particular importance, we will omit the phrase “of degree τ”. Recall that according to Definition 2.7, two CCGPSR manifolds of the same degree are called equivalent if they are related by a linear change of coordinates of the ambient space. Also note that CCGPSR manifolds are automatically maximal in the sense of Definition 2.9.
Lemma 2.11**.**
Let H⊂{h=1}⊂Rn+1 be an n-dimensional GPSR manifold of degree τ≥3. Then its centro-affine fundamental form gH is given by
[TABLE]
where ∂2 is determined by the chosen linear coordinates on the ambient space Rn+1.
When studying CCGPSR manifolds of degree τ≥3 with the aim of some kind of classification, it is useful to introduce the notion of their respective moduli sets.
Definition 2.12**.**
Let n∈N∪{0}. We define the moduli set of n-dimensional CCGPSR manifolds of degree τ to be the set of equivalence classes
[TABLE]
where [H]=[H] if and only if H and H are equivalent. For τ=3, we will call the above set the moduli set of n-dimensional CCPSR manifolds.
An important topological property of the cone spanned by CCGPSR manifolds which we will need in our studies is its convexity:
Proposition 2.13**.**
Let H⊂{h=1}⊂Rn+1 be an n-dimensional CCGPSR manifold. Then
[TABLE]
is a convex cone
and the map R>0×H∋(r,p)↦r⋅p∈U
is a diffeomorphism.
Proof.
[CNS, Prop. 1.10] for the special case of CCGPSR manifolds.
∎
We will later parametrise CCGPSR manifolds over sections of their respective spanned cone with an affinely embedded tangent space, cf. equation (3.18). A key fact that we will need in order to prove the compactness of the set Cn in Theorem 1.1 is the precompactness of these sections, which is part of the following lemma:
Lemma 2.14**.**
Let H be a CCGPSR manifold and let U=R>0⋅H. Then for every p∈H, the intersection
[TABLE]
is open, precompact, and convex. Here (p+TpH)⊂Rn+1 denotes the affinely embedded tangent space TpH in the ambient vector space Rn+1 equipped with the induced subspace topology.
Proof.
[CNS, Lem. 1.14] applied to homogeneous polynomials.
∎
In order to further specify types of CCGPSR manifolds we define a certain type of boundary behaviour:
Definition 2.15**.**
Let H⊂{h=1}⊂Rn+1 be a CCGPSR manifold and let U=R>0⋅H be the corresponding convex cone. We will call H singular at infinity if there exists a point p∈∂U∖{0}, such that dhp=0.
Definition 2.15 is not empty for CCPSR manifolds in the sense that for each n≥1, there exists an n-dimensional CCPSR manifold that is singular at infinity. This is a consequence of Proposition 4.6 and the existence of homogeneous CCPSR manifolds in all dimensions, cf. [CHM] for dimension 1, [CDL] for dimension 2, and [DV] for dimension n≥3. It will turn out that being singular at infinity or not already determines the regularity of the boundary behaviour of CCPSR manifolds in the sense of Definition 4.10, for the result see Theorem 4.12.
Remark 2.16**.**
A natural question that arises when studying PSR manifolds is whether it is possible to classify all closed connected PSR manifolds up to equivalence. In general, this turns out to be a very difficult question. This problem is equivalent to classifying all cubic hyperbolic homogeneous polynomials up to equivalence. One of the encountered difficulties is that being hyperbolic for a cubic homogeneous polynomial is an open condition in the sense that if h∈Sym3(Rn+1)∗ is hyperbolic and H∈Sym3(Rn+1)∗ is any cubic polynomial, then there exists an ε>0, such that for all 0≤k≤ε the polynomial h+kH is hyperbolic. This follows easily from Sylvester’s law of inertia. Furthermore, the dimension of Sym3(Rn+1)∗ grows cubically in n while the dimension of GL(n+1) grows quadratically in n, so we can not expect to have only finitely many examples as n grows large. In dimensions n=1 and n=2 however, cubic hyperbolic homogeneous polynomials in 2 and 3 variables, respectively, and the corresponding closed connected PSR manifolds have been classified up to equivalence, see [CHM] for 1-dimensional PSR manifolds and [CDL] for 2-dimensional PSR manifolds. Aside from the low-dimensional restriction, another restriction to PSR manifolds is to consider only those that are contained in the level set of a reducible cubic hyperbolic homogeneous polynomial. In this case, CCPSR manifolds are classified in any dimension [CDJL, Thm. 2]. Lastly, there is a classification of PSR manifolds that are homogeneous spaces under the action of their respective automorphism groups, cf. [DV].
In this work we will use the classification result of CCPSR surfaces:
Let H⊂R3 be a CCPSR surface. Then H is equivalent to exactly one of the following:
a)
{xyz=1,x>0,y>0},
2. b)
{x(xy−z2)=1,x>0},
3. c)
{x(yz+x2)=1,x<0,y>0},
4. d)
{z(x2+y2−z2)=1,z<0},
5. e)
{x(y2−z2)+y3=1,y<0,x>0},
6. f)
{y2z−4x3+3xz2+bz3=1,z<0,2x>z}* for precisely one b∈(−1,1).*
3 Standard form for GPSR manifolds and their curvature tensors
In this section we are going to develop the technical tools necessary to prove our results. The first and maybe central one is the existence of a certain standard form of GPSR manifolds in dependence of a chosen reference point and which, at least locally, varies smoothly along said reference point.
Proposition 3.1**.**
Let H⊂{h=1}⊂Rn+1 be an n≥1-dimensional connected GPSR manifold of degree τ≥3. Then for each p∈H there exists a linear change of coordinates on Rn+1 described by A(p)∈GL(n+1), such that
(i)
(h∘A(p))((xy))=xτ−xτ−2⟨y,y⟩+k=3∑τxτ−kPk(y),
2. (ii)
A(p)⋅(10)=p,
where y=(y1,…,yn)T denote the standard linear coordinates of Rn, (xy) denotes the corresponding coordinates of Rn+1≅R×Rn, (10) denotes the point (xy)=(10)∈Rn+1, and ⟨⋅,⋅⟩ denotes the standard Euclidean scalar product on Rn induced by the y-coordinates. Furthermore, if H is a CCGPSR manifold then the transformations A(p) can be chosen in such a way that A:H→GL(n+1) is smooth. If H is not closed as a subset of Rn+1, we can still find for each p∈H a subset V⊂H that contains p and is open in the subspace-topology of H⊂Rn+1, such that A:V→GL(n+1) can be chosen so that it is a smooth map.
Proof.
First we will show that (i) and (ii) hold for all connected GPSR manifolds. Then we will prove that in the case of CCGPSR manifolds, A:H→GL(n+1) can be chosen to be smooth. In the case of connected GPSR manifolds which are not necessarily closed we will show that for all p∈H there always exists an open neighbourhood V⊂H of p, and that A:V→GL(n+1) can be chosen so that it is a smooth map.
Let H⊂Rn+1 be a connected GPSR manifold and denote by ⟨⋅,⋅⟩ the standard Euclidean scalar product on Rn+1 induced by the choice of the linear coordinates on Rn+1. Let p∈H be arbitrary. We will differentiate between two cases.
Case 1: dhp=r⟨p,⋅⟩ for some r=0.
Note that the property dhp∈(R∖{0})⋅⟨r,⋅⟩ is preserved by changing the linear coordinates of the ambient space Rn+1 by rotations in SO(n+1) and by positive rescaling of the linear coordinates. We can thus without loss of generality assume that p=(1,0,…,0)T,
and denote the linear coordinates on Rn+1 by (x,y1,…,yn)T. Since h(p)=1 is a necessary condition for p∈H, we find that h must be of the form
[TABLE]
where L∈Lin(Rn,R) is linear in y and Q∈Sym2(Rn)∗ is a symmetric bilinear form. We can now check that dhp∈(R∖{0})⋅⟨r,⋅⟩ implies L≡0. By assumption, p is a hyperbolic point of H. We calculate
[TABLE]
The hyperbolicity of the point p thus shows that Q must be negative definite. Hence, after a suitable transformation of the y-coordinates, we find that h can be transformed into the desired form
[TABLE]
Case 2: dhp=r⟨p,⋅⟩ for all r=0.
Note that in this case, r=0 is automatically excluded by dhp(p)=τ=0. We will find a linear coordinate transformation B∈GL(n+1) of the ambient space Rn+1 of H, such that Bq=p and
[TABLE]
which will take us to the setting of the first case since d(h∘B)q=dhBq(B⋅). Note that in the above equation (3.1), ⟨⋅,⋅⟩ denotes the Euclidean scalar product induced by the new coordinates, that is the standard linear coordinates in the domain of B:Rn+1→Rn+1. In order to prove the existence of such a transformation B, let
[TABLE]
We claim that ⟨⟨⋅,⋅⟩⟩>0. To show this, write v∈Rn+1∖{0} as v=ap+w, w∈p⊥⟨⋅,⋅⟩.
Note that a and w are uniquely determined since Rn+1=Rp⊕p⊥⟨⋅,⋅⟩. We obtain
[TABLE]
For w=0 we immediately see that ⟨⟨v,v⟩⟩>0. For w=0, v=0 implies a=0. In that case ⟨⟨v,v⟩⟩=a2τ2>0. This shows that ⟨⟨⋅,⋅⟩⟩ is indeed positive definite. Now let B∈GL(n+1) be an orthonormal basis111We interpret the columns of B as the basis vectors. of ⟨⟨⋅,⋅⟩⟩, that is B∗⟨⟨⋅,⋅⟩⟩=⟨⋅,⋅⟩.
Denote by h=h∘B the transformed polynomial h and let q=B−1p. Then dhq=dhBq(B⋅)=dhp(B⋅)
and
[TABLE]
Hence, B fulfils (3.1) with r=τ1 and we have dhq=τ1⟨q,⋅⟩ with q∈B−1H. We are now in the setting of the first case and can proceed as described therein.
Summarising up to this point, we have shown that for any n≥1-dimensional connected GPSR manifold H⊂{h=1} and all p∈H we can find A∈GL(n+1), such that the conditions (i) and (ii) are fulfilled. Now we will describe how to construct A explicitly.
We will start with the case where H is a CCGPSR manifold, and first construct the transformation A(p) explicitly for one arbitrarily chosen point p∈H, so that A(p) fulfils (i) and (ii). We start by choosing initial linear coordinates (x,y1,…,yn)T of Rn+1 and a point p=(pxpy)∈H. After a possible reordering of the coordinates we can assume that ∂x∂h(p)=0. This follows from dhp=0, since otherwise τh(p)=dhp(p)=0. Let
[TABLE]
where ∂xh:=∂x∂h and ∂yh:=i=1∑ndh(∂yi)dyi. A∈GL(n+1) follows from
[TABLE]
In the above formula we have used the Euler identity for homogeneous functions. We find A⋅(10)=p and
[TABLE]
The vanishing of the xτ−1-term follows from dhp(−∂xh∂yhp(y)y)=0 for all y∈Rn. This is equivalent to (−∂xh∂yhp(y)y)∈TpH for all y∈Rn. Hence,
[TABLE]
is a positive definite bilinear form since p is, by assumption, a hyperbolic point of h. This implies that there exists a linear transformation E∈GL(n), such that
[TABLE]
Since \widetilde{A}\cdot\left(\begin{tabular}[]{c|c}1&\\
\hline\cr&\overset{}{\widetilde{E}}\end{tabular}\right)\cdot\left(\begin{matrix}1\\
0\end{matrix}\right)=p, we have shown that for one choice of p∈H we can find a linear transformation fulfilling both (i) and (ii).
In order to prove the statement of this proposition for all p∈H, we have shown that we can assume without loss of generality that h is of the form (i) and that (10)∈H⊂{h=1}. For p=(pxpy)∈H and E(p)∈GL(n) consider the matrix
[TABLE]
Firstly we need to ensure that A(p) is well-defined for all p∈H and all choices for E(p)∈GL(n). This follows from
[TABLE]
which we will prove next. In order to show that (3.7) holds for all n≥1-dimensional CCGPSR manifolds, it in facts suffices to prove it for all 1-dimensional CCGPSR manifolds. To see this, suppose that dim(H)>1 and that there exists a point p=(pxpy)∈H, such that ∂xh∣p=0. Then the set
[TABLE]
is a 1-dimensional CCGPSR manifold which coincides with the connected component of the level set
[TABLE]
that contains the point (10)∈R2. In (3.8), v∈Rn is chosen to fulfil span{(10),p}=span{(10),(0v)} and ⟨v,v⟩=1. Note that h:=h(x(10)+y(0v)) is then automatically of the form (i). Denote by p=(pxpy)∈R2 the point fulfilling px(10)+py(0v)=p and note that py=0. Then p∈H by construction and ∂xh∣p=0. It now follows from Lemma 2.14 that there exists R>0, such that h(p+R(10))=0, since (10)∈TpH by assumption. The convexity of the cone U:=R>0⋅H⊂R2 (cf. Proposition 2.13) implies that
[TABLE]
But p∈V, and we conclude with H⊂U that p∈H, which is a contradiction. We have thus shown that (3.7) holds for every n≥1-dimensional CCGPSR manifold H.
We now show that for all p∈H and all choices for E(p)∈GL(n), A(p)∈GL(n+1). The calculation is similar to calculating det(A) (3.4) and yields
[TABLE]
In order to obtain the conditions for E(p) so that A(p) fulfils condition (i), we calculate
[TABLE]
By definition, dhp(−∂xh∂yhp(E(p)y)E(p)y)=0 for all y∈Rn and all E(p)∈GL(n), which is equivalent to (−∂xh∂yhp(E(p)y)E(p)y)∈TpH for all y∈Rn and all choices E(p)∈GL(n). Thus,
the bilinear form in equation (3.5) is a positive definite bilinear form
since H⊂{h=1} consists only of hyperbolic points of the defining polynomial h. We conclude that for all p∈H, E(p)∈GL(n) can be chosen in such a way that
[TABLE]
for all y∈Rn.
Summarising, we have demonstrated for each p∈H how to explicitly construct a linear change of coordinates A(p)∈GL(n+1) which fulfils (i) and (ii). It remains to show that the assignment A:H→GL(n+1) can be chosen so that it is a smooth map. To see this observe that
[TABLE]
The matrix \left(\begin{tabular}[]{c|c}p_{x}&-\left.\frac{\partial_{y}h}{\partial_{x}h}\right|{p}\\
\hline\crp{y}&\overset{}{\mathbbm{1}}\end{tabular}\right) in the above equation depends smoothly on p∈H. Hence, it suffices to show that E:H→GL(n) can be chosen so that it is a smooth map and fulfils equation (3.9). This follows from the fact that, as we have seen above,
[TABLE]
understood as in (3.5) is positive definite for all p∈H, cf. [Le, Lem. 8.13].
It remains to deal with the cases where H⊂{h=1}⊂Rn+1 is a connected GPSR manifold, but is not closed in Rn+1. For p∈H arbitrary and fixed, we want to show that there exists a neighbourhood V⊂H of p in H, such that A:V→GL(n+1) can be chosen to fulfil (i) and (ii) and to be a smooth map. We have already seen in the beginning of the proof that we can, after a possible linear transformation of the coordinates of Rn+1, assume without loss of generality that p=(10), that h is of the form h=xτ−xτ−2⟨y,y⟩+k=3∑τxτ−kPk(y), and that H is the contained in the connected component of {h=1} that contains the point (10)∈H. Since ∂xh∣(10)=τ>0, it immediately follows that we can find a neighbourhood V of (10) in H, such that ∂xh∣q>0 for all q∈V. We can now define A as in equation (3.6) and proceed as for the case when H was assumed to be closed.
∎*
*
Proposition 3.1 shows in particular that for any CCGPSR manifold H⊂{h=1} we can assume without loss of generality that h is of the form
[TABLE]
and that H is the precisely the connected component of {h=1} which contains the point (xy)=(10)∈Rn+1. If H is just assumed to be a connected an not necessarily closed GPSR manifold, we can still assume without loss of generality that H is a connected open subset of {h=1} with h of the form (3.10), and that H contains the point (10)∈Rn+1. Also note that whenever H is a CCGPSR manifold, the point (10)∈H is the unique point in H that minimises the Euclidean distance of H⊂Rn+1 and the origin 0∈Rn+1 (in the chosen linear coordinates (xy) of Rn+1).
Further observe that for connected PSR manifolds the term P3 is never uniquely determined. To see this consider H⊂{h=1} with h of the form (3.10). If P3=0, the transformation y↦−y will preserve the form (3.10) and send P3↦−P3. For P3=0, one can verify that for any point p∈H, p=(10), the corresponding coordinate transformation A(p) of the form (3.6) will induce a non-zero P3-part in the transformed polynomial h.
Proposition 3.1 is also useful to unclutter the rest of our studies by introducing the term standard form of GPSR manifolds as follows:
Notation**.**
The statement that a GPSR manifold H is in standard form will in the following mean that
•
H⊂{h=1}⊂Rn+1 is an n≥1-dimensional GPSR manifold of degree τ≥3,
•
we have chosen linear coordinates (xy)=(x,y1,…,yn)T on the ambient space Rn+1 of H, such that h is of the form (3.10) and (xy)=(10)∈H.
By Proposition 3.1 we know that assuming that a GPSR manifold is in standard form is not a restriction of generality. We might further specify the degree τ of H or impose topological properties such as maximality or the dimension, which we will then denote by e.g. “let H be a maximal PSR manifold of dimension n≥3 in standard form”. Using the abbreviation “standard form” will thus allow us to omit stating every time that the defining polynomial of a GPSR manifold H is assumed to be of the form (3.10), the dimension of H has to be ≥1, and that for the linear coordinates (xy) of the ambient space the point (10) is assumed to be an element of H. This will make the following statements considerably easier to read. Occasional, however, we might write out additional information that is already implied by the term “standard form” in order to make specific statements easier to understand, see e.g. Corollary 4.13.
Next, we will calculate standard forms of CCPSR surfaces, cf. Theorem 2.17. Aside from serving as examples for the techniques developed in Proposition 3.1, the following calculations will be important in proving the later Theorem 4.15 which is one of the main ingredients we need to prove our main Theorem 1.1.
Example 3.2**.**
Let (x,y,z)T denote the standard linear coordinates on R3. Recall that CCPSR surfaces H⊂{h=1}⊂R3 have been classified up to equivalence in [CDL, Thm. 1], cf. Theorem 2.17 a)–f). In the following we will for each h corresponding to the cases a)–f) give a choice of A=A(p)∈GL(3) corresponding to a given point p∈H, such that A⋅(100)=p, h(A⋅(xyz)) is of the form (3.10), and A−1(H)⊂{h∘A=1} is precisely the connected component of {h∘A=1}⊂R3 that contains the point (xyz)=(100).
a) H={h=xyz=1,x>0,y>0}.
It is clear that p=(1,1,1)T∈H. One choice for the corresponding linear transformation of the form (3.6) is
[TABLE]
which brings h to the form
[TABLE]
b) H={h=x(xy−z2)=1,x>0}.
Similar to the surface in a), consider the point p=(1,1,0)T∈H and
[TABLE]
Then
[TABLE]
c) H={h=x(yz+x2)=1,x<0,y>0}.
With p=(−1,2,−2)T∈H and
[TABLE]
we obtain
[TABLE]
d) H={h=z(x2+y2−z2)=1,z<0}.
By re-ordering of the coordinates and switching one sign one quickly finds that H is equivalent to H={h=x3−x(y2+z2)=1,x>0}, which is precisely the connected component of {h=1} that contains the point (x,y,z)T=(1,0,0)T. The corresponding point in H and transformation A∈GL(3) are given by p=(0,0,1)T∈H and
[TABLE]
so that indeed
[TABLE]
The transformation A is not of the form (3.6) since we needed to switch the x- and z-coordinate so that ∂x(h∘A)∣p=0.
e) H={h=x(y2−z2)+y3=1,y<0,x>0}.
Consider the point p=(2,−1,0)T∈H and the corresponding linear transformation as in (3.6)
[TABLE]
We find
[TABLE]
f) Hb={h=y2z−4x3+3xz2+bz3=1,z<0,2x>z}, b∈(−1,1).
Observe that the point pb=31−b1(1/2,0,−1)T is contained in Hb for all b∈(−1,1). After switching the x- and z-coordinate via the transformation (111), we can apply the construction in equation (3.6) in order to find Ab∈GL(3), such that h∘((111)⋅Ab) is of the form (3.10). We find
[TABLE]
With Ab:=(111)⋅Ab
we obtain
[TABLE]
and have thus shown that h∘Ab is of the form (3.10) and that Ab⋅(1,0,0)T=pb for all b∈(−1,1) as required. Note that equation (3.16) allows us to interpret the one-parameter family of CCPSR surfaces Hb as an interpolation between the CCPSR surfaces Theorem 2.17 d) (for b→1, see equation (3.14)) and Theorem 2.17 e) (for b→−1). To see the latter, observe that with
[TABLE]
the polynomial
[TABLE]
which is precisely the limit polynomial of (3.16) for b→−1 after to swapping y and z, transforms to
[TABLE]
which coincides with equation (3.15). Furthermore one can check that the point A⋅(1,0,0)T is contained in the connected component of {x3−x(y2+z2)+332y3−231yz2=1} that contains the point (x,y,z)T=(1,0,0)T, for which we have shown that this is equivalent to the CCPSR surface e) in Theorem 2.17. Hence, the connected component of {x3−x(y2+z2)+332y3=1}
that contains the point (x,y,z)T=(1,0,0)T is in particular also a CCPSR surface which is equivalent to the surface e).222In order to find the transformation A, we have used a technique developed later in Theorem 4.15. Specifically we used equations (4.27) and (4.28).
For the following considerations it is helpful to consider a certain parametrisation of connected GPSR manifolds which we will introduce next.
Definition 3.3**.**
Let H be a connected GPSR manifold in standard form.
We define
[TABLE]
where prRn:Rn+1→Rn, (xy)↦y.
The set dom(H) is precisely the intersection of the cone spanned by H, that is R>0⋅H⊂Rn+1, and T(10)H affinely embedded in Rn+1 via v↦(10)+(0v).
Independent of whether the connected GPSR manifold H⊂{h=1}⊂Rn+1 is closed or not, dom(H)⊂Rn is well-defined, open in Rn, and always contains an open ball Bε(0)⊂Rn with respect to the standard scalar product ⟨⋅,⋅⟩ on Rn for ε>0 small enough. In order to check that these claims are true, one uses the following facts. Firstly, every ray R>0⋅p for p∈H meets H precisely once. This follows from the homogeneity of degree τ≥3 of the corresponding polynomial h:Rn+1→R. Secondly, H⊂{x≥1}⊂Rn+1 and H∩{x=1}=(10). This follows from the fact that H is locally around each point in H contained in the boundary of a strictly convex domain of in Rn+1, which in turn follows from the Sacksteder-van Heijenoort Theorem333To apply said theorem, one first needs to extend the considered local neighbourhood of H to a Euclidean complete convex hypersurface.
[Wu]. Note that if H is a CCGPSR manifold, then H is (globally) the boundary of the strictly convex domain R>1⋅H⊂Rn+1.
Thus, every ray R>0⋅p for p∈H has a unique intersection-point with the set dom(H). We see that dom(H) is
bijective to H via
[TABLE]
One can check that Φ is everywhere a local diffeomorphism. This and H being a hypersurface of Rn+1 also show that dom(H)⊂Rn is open and, hence, that Φ is a diffeomorphism444This is the reason behind our choice of the term “dom”, which is to be thought of as an abbreviation of “domain”..
Note, however, that the set dom(H) does depend on the chosen linear coordinates of the ambient space Rn+1.
Lemma 2.14 implies the following property of dom(H) if H is a CCGPSR manifold.
Corollary 3.4**.**
Let H be a CCGPSR manifold in standard form. Then dom(H)⊂Rn is open, precompact, and convex.
The statement of Corollary 3.4 is in particular independent of the linear coordinates of the ambient space Rn+1 of H.
We will use the parametrisation (3.18) of H⊂{h=1} to study infinitesimal changes of the Pk’s in the standard form (3.10) of h when we vary the point p∈H in Prop. 3.1 (i) near (xy)=(10)∈H. The results are important technical tools for our following studies.
Whenever we use z-variables from here on, we will be working with dom(H). The y-variables will be used in when working with the ambient space Rn+1 of H.
For the following calculations we will define the (globally smooth) functions
[TABLE]
Note that whenever H is closed and connected, dom(H) coincides with the connected component of {β(z)>0} that contains the point z=0∈Rn, and β∣∂dom(H)≡0. Also, as shown in the proof of Proposition 3.1, α∣dom(H)>0 if H is a CCGPSR manifold. If H is not closed, we can at least find a neighbourhood V of z=0∈Rn, such that α∣V>0, which also follows from the proof of Proposition 3.1. Furthermore, it immediately follows from (3.18) that Φ(z)=τβ(z)1(1z) for z∈dom(H). While dh does not vanish on H, it might vanish at a point (xy)=(1z) for z∈∂dom(H) or, equivalently, on the ray R>0⋅(1z)⊂∂(R>0⋅H). If H is furthermore closed, we are in this case precisely in the setting of CCGPSR manifolds that are singular at infinity, cf. Definition 2.15. The following lemma characterises these cases for CCGPSR manifolds in terms of the functions α and β.
Lemma 3.5**.**
Let H be a CCGPSR manifold in standard form and
let α, β be defined as in (3.19), respectively (3.20). Then for all z∈∂dom(H) the following are equivalent:
(i)
dh(1z)=0,
2. (ii)
α(z)=0,
3. (iii)
dβz=0.
Proof.
Assume that dh(1z)=0 for a z∈∂dom(H). By affinely embedding dom(H) into Rn+1 via z↦(1z) and identifying y and z we obtain
[TABLE]
Since α(z)dx and dβz are linearly independent we conclude that α(z)=0 and dβz=0.
Now assume that dβz=0. Then, using the Euler-identity for homogeneous functions, 0=τβ(z)=dh(1z)(1z)=α(z)
showing that α(z)=0. Hence, dh(1z)=0.
Lastly, assume that α(z)=0. Similar to above, 0=τβ(z)=dh(1z)(1z)=dβz(z). We need to show that this implies dβz=0. Assume the latter does not hold. Then dh(1z)=0 and, hence, we can use the implicit function theorem and conclude that dom(H) has smooth boundary near z, and dβz(z)=0 is equivalent to the statement that z∈Tz∂dom(H). This, however, contradicts the assumption that H is a CCGPSR manifold which implies that dom(H) is a convex set containing the point 0∈Rn (cf. Lemma 2.14). To see the contradiction, observe that for each non-singular point z∈∂dom(H), i.e. a point satisfying dβz=0, the affinely embedded tangent space z+Tz∂dom(H) in Rn intersects the convex compact set dom(H) (cf. Corollary 3.4) only at its boundary, that is ∂dom(H). But if z∈Tz∂dom(H), the intersection of z+Tz∂dom(H) and dom(H) will always contain 0∈Rn which is, independently of any coordinate choice of the ambient space Rn+1 of H, always contained in dom(H) and in particular never contained in ∂dom(H). This follows directly from the definition of dom(H), see Definition 3.3. This is a contradiction to the convexity of dom(H), see Corollary 3.4.
∎
Using Proposition 3.1, we will now study the infinitesimal changes in the transformations A(p) for p∈H near (xy)=(10)∈H, and in the corresponding polynomials Pi in the considered polynomial h as in equation (3.10). To do so we use the parametrisation Φ:dom(H)→H given in equation (3.18).
With this in mind the next result is an infinitesimal analogue of Proposition 3.1 and has applications in e.g. significantly simplifying the calculation of the first derivative of the scalar curvature of CCPSR manifolds, cf. Proposition 3.12.
Proposition 3.6**.**
Let H be a connected GPSR manifold in standard form and
let V⊂H be an open neighbourhood of (xy)=(10) and A:V→GL(n+1),
[TABLE]
as in equation (3.6) so that A(p) fulfils (i) and (ii) in Proposition 3.1 with A((10))=\mathbbm1. Let Φ:dom(H)→H be the diffeomorphism given in equation (3.18) and define
[TABLE]
Then there exists an so(n)-valued linear map dB0∈Lin(Rn,so(n)) of the form
[TABLE]
where {ak∣1≤k≤n(n−1)/2} is a basis of so(n) and ℓk∈Rn for all 1≤k≤n(n−1)/2,
such that for τ≥4
[TABLE]
and for τ=3, that is for connected PSR manifolds,
[TABLE]
In the above equations, P3(y,⋅,dz)T is to be understood as the column-vector 61∂2P3∣y((dz1⋮dzn),⋅)T
and dB0 is to be understood as
[TABLE]
Proof.
Note that z=0∈Φ−1(V) for all possible choices for V since Φ−1((10))=0. Observe that for all v∈Rn, the function −∂xh∂yh(v) defined on R>0⋅H is constant along rays of the form R>0⋅p, p∈H. With the notation E(z)=E(Φ(z)) and α, β defined in (3.19), respectively (3.20),
[TABLE]
and A(0)=\mathbbm1, E(0)=\mathbbm1, α(0)=τ, ∂2β0=−2⟨dz,dz⟩.
We obtain
[TABLE]
where we understand dz as the identity-map on Rn and dE0 as a gl(n)-valued 1-form, dE0∈Ω1(Rn,gl(n)), both using the identification T0dom(H)≅Rn obtained with the affine embedding dΦ0 as in equation (3.18). With
[TABLE]
we get
[TABLE]
The assumption that A fulfils (i) and (ii) in Proposition (3.1) and A(0)=\mathbbm1 implies that the xτ−2-term in the above equation (3.25) must vanish, i.e. −2⟨y,dE0y⟩+3P3(y,y,dz)=0.
This is true if and only if
[TABLE]
for all y∈Rn, with dB0∈Lin(Rn,so(n)) a linear map from Rn to so(n). Here we have identified Rn with T0dom(H).
We will now justify our notation of the endomorphism dB0. Consider for any smooth map B:Rn→O(n) with B(0)=\mathbbm1 the map
[TABLE]
It is clear that if we replace E with (B∘Φ−1)⋅E in the map A (and correspondingly B⋅E in A), it will still fulfil (i) and (ii) in Proposition 3.1 and A(0)=\mathbbm1. We can thus choose for any dB0∈Ω1(Rn,so(n)) a fitting map B:Rn→O(n) and a smooth map E:Φ−1(V)→GL(n) with dE0=23P3(y,⋅,dz)T, E(0)=\mathbbm1, so that E:=B⋅E will fulfil equation (3.26). Also note that the requirement B(0)=\mathbbm1 implies that the image of B lies in SO(n).
To complete the proof, we only need to replace dE0 in dh(xy)(dA0(xy)) as in equation (3.26) and obtain the claimed result.
∎
Equation (3.22) in Proposition 3.6 determines precisely the infinitesimal changes of the Pk’s in the polynomial h as in equation (3.10) when changing coordinates for p∈H⊂{h=1} parametrised by Φ:dom(H)→H (3.18) in the way described by Proposition 3.1. Rotations in y∈Rn⊂Rn+1 always preserve (3.10), which is seen in the freedom of choosing dB0∈Lin(Rn,so(n)). We will now assign symbols to the respective infinitesimal changes of the Pk’s in order to simplify the considerations to follow.
Definition 3.7**.**
With the assumptions of Proposition 3.6 and the definition of A as in (3.21), we define for τ≥3 and 3≤k≤τ
[TABLE]
where we denote by ∂z∂=i=1∑ndzi⊗∂zi the de-Rham differential with respect to the z=(z1,…,zn)T-coordinates. In particular, we have for τ=3, that is cubic polynomials h,
[TABLE]
and for τ=4, that is quartic polynomials h,
[TABLE]
This means that the δPk(y)’s are precisely the factors depending on y in the summands of
[TABLE]
that are of order xτ−k, respectively. For each 3≤k≤τ we call δPk the first variation of Pk along H with respect to the chosen dB0 (3.26), respectively dA0 (3.24), and understand δPk(y) as a linear map δPk(y):Rn→Symk(Rn)∗, so that we insert vectors v∈Rn into the dz in each δPk(y) and obtain a homogeneous polynomial in (y1,…,yn) of degree k.
The first application for Proposition 3.6 that we will consider is calculating the first derivative of the scalar curvature of a connected GPSR manifold H equipped with its centro-affine fundamental form at one certain point. To do so we need a closed form of the scalar curvature (at at least one point). Its calculation uses the following result.
Lemma 3.8**.**
Let H be a connected GPSR manifold of degree τ≥3 in standard form with centro-affine fundamental form gH=−τ1∂2h∣TH×TH (cf. Lemma 2.11) and
let Φ:dom(H)→H be the diffeomorphism given in equation (3.18) and β as in equation (3.20). Then
[TABLE]
Proof.
This is a special case of [CNS, Cor. 1.13]. To check the claim, one uses the homogeneity of degree τ−2≥1 of ∂2hp in p and the first derivative of the diffeomorphism Φ:dom(H)→H (3.18), that is
[TABLE]
∎
We will use equation (3.29) to calculate the scalar curvature of (dom(H),Φ∗gH) at z=0∈dom(H).
Proposition 3.9**.**
Let H be an n≥2-dimensional connected GPSR manifold of degree τ≥3 in standard form.
Then the scalar curvature SH:H→R of (H,gH) at the point (10)∈H is given by
[TABLE]
where ∂k=∂yk for 1≤k≤n.
Proof.
We identify ∂zi=∂yi=∂i when inserting vectors in the polynomials Pk. This is justified by the fact that dΦ0 bijectively maps T0dom(H) to T(10)H via dΦ0:∂zi↦∂yi for all 1≤i≤n.
For the following calculations we will first calculate the scalar curvature S:dom(H)→R of (dom(H),g:=τΦ∗gH) at z=0. We work with g instead of Φ∗gH because the necessary calculations will then require less symbols. Furthermore, we will for the general calculations assume that τ≥4. The calculations for τ=3 are analogous.
From equation (3.29) it follows that g is given by
[TABLE]
We abbreviate ∂zμ=∂μ and obtain for the first entry-wise derivative of g in zμ-direction
[TABLE]
The second partial derivatives of g read
[TABLE]
Applying the above formulas at z=0, we obtain
[TABLE]
In order to calculate the scalar curvature of (dom(H),g)≅(H,τgH),
[TABLE]
at z=0, we need to calculate the Christoffel symbols and their first derivatives at that point. We have
[TABLE]
and
[TABLE]
Since gij0=21δij,
[TABLE]
We obtain
[TABLE]
Hence,
[TABLE]
Recall that dΦ0(∂zi)=∂yi for all 1≤i≤n, which one can easily verify. Thus SH((10))=τS(0) together with the above equation prove our claim. Observe that SH((10)) only depends on the dimension n of H, the degree of homogeneity τ, and the cubic polynomial P3. Also note that SH≡0 for dim(H)=n=1 is consistent with formula (3.30).
∎
Proposition 3.9 gives us, at least in theory, a simple way of calculating the scalar curvature of a connected GPSR manifold H equipped with its centro-affine fundamental form (and thus of all GPSR manifolds by considering each connected component) at every point p∈H. This, however, requires calculating A(p) as in Proposition 3.1 for each p∈H. This amounts basically to determining an orthonormal basis for a positive definite bilinear form depending on p∈H. This is certainly easier than calculating Christoffel-symbols and their derivatives at each point, but nevertheless complicated enough to require a (both p- and H-dependent) case-by-case study and not giving us a closed form of SH(p) for all p∈H.
Calculating the first derivative of the scalar curvature SH at the point (xy)=(10)∈H can of course also be done in a direct way, but the calculations require the (local) calculation of the third partial derivatives of the metric gH and, hence, are very long and have a huge potential for human error. One can however also make use of Proposition 3.6 to obtain a formula for dSH∣(10).
Proposition 3.10**.**
With the assumptions and notations of Proposition 3.9 and Definition 3.7 and identifying T(10)H with the affinely embedded hyperplane {(0y)y∈Rn}⊂Rn+1, we have for τ≥4
[TABLE]
and for τ=3
[TABLE]
Proof.
In the following calculations we will identify dz and dy, respectively each ∂zi and ∂yi (and write ∂i instead) via dΦ0, cf. equation (3.18), which has the property that dΦ0(∂zi)=∂yi for all 1≤i≤n. We start with the case τ≥4. With the notations of Definition 3.7, Propositions 3.9 and Proposition 3.6 equation (3.22) imply
[TABLE]
where
[TABLE]
Recall that dB0∈Lin(Rn,so(n)). We thus need to determine a formula for δP3(∂i,∂j,∂k) for all 1≤i,j,k≤n. The safest way in the sense that possible errors in the pre-factors do not occur is to determine ∂2(δP3(y)), where we regard dz in equation (3.36) as a constant vector. We obtain
[TABLE]
for all y,v∈Rn and, hence,
[TABLE]
for all y,v,w∈Rn. Since δP3(y) is homogeneous of degree 3 in y, we have the identities
[TABLE]
when we regard dz as a constant vector and interpret δP3 as a cubic tensor. We use the above identities and obtain
[TABLE]
and
[TABLE]
To see that all terms containing dB0:Rn→so(n) (understood as in equation (3.23)) vanish, observe that for all 1≤a,i,ℓ≤n the tensors
[TABLE]
are symmetric in their two arguments. Their trace with respect to the standard Euclidean scalar product ⟨⋅,⋅⟩ on Rn when inserting any matrix M∈so(n) in one of the arguments thus vanishes.
We can now use the above formulas for a,i,ℓ∑P3(∂a,∂a,∂ℓ)δP3(∂i,∂i,∂ℓ) and a,i,ℓ∑P3(∂a,∂i,∂ℓ)δP3(∂a,∂i,∂ℓ) in equation (3.35) and, with the identification of dz and dy via dΦ0 (3.18), obtain our claimed result for τ≥4. For τ=3, observe that the formulas for δP3 in equations (3.28) and (3.36) coincide when setting P4≡0. The calculations for the case τ=3 thus coincide with the cases τ≥4 and we obtain the claimed result.
∎
The calculations used in Proposition 3.9 can also be used to calculate the Riemannian curvature tensor, the Ricci curvature, and the sectional curvatures of a connected GPSR manifold (H,gH).
Lemma 3.11**.**
With the assumptions of Proposition 3.9, let R denote the Riemannian curvature tensor, Ric denote the Ricci curvature, and K denote the sectional curvature of an n-dimensional connected GPSR manifold (H,gH), respectively. We again identify dz and dy at (10)∈H via dΦ0 (3.18). Then
[TABLE]
and for dim(span{v,w})=2
[TABLE]
where F∈O(n) is any orthogonal transformation with the property that span{v,w}=span{F∂i,F∂j}. Note that such a transformation F always exists for any choices of i=j, and that K(v,w) does in particular not depend on that choice of i, j, and the corresponding F.
Proof.
The formulas (3.37) and (3.38) for the Riemannian curvature tensor R and the Ricci tensor Ric, respectively, follow directly from the formulas for the Christoffel symbols (3.33), their first derivatives (3.34), and the inverse of gH at the point (xy)=(10) (3.32) (up to the factor τ) given in the proof of Proposition 3.9. Recall that in said proof we work with g=τΦ∗gH, Φ as in (3.18), hence we also need to rescale the formula for g at [math] (3.32) at the point where we take the trace with respect gH. For the sectional curvature K, the formula for K(10)(∂i,∂j) for i=j follows easily from (3.37) and (3.31) (and by rescaling with the overall factor τ). To find the general formula K(10)(v,w) (3.39) for any two linearly independent vectors v,w∈T(10)H≅Rn, choose i=j and F∈O(n) as described such that span{v,w}=span{F∂i,F∂j}. Changing the coordinates of the ambient Rn+1 via
[TABLE]
corresponds to rotating H in the y-coordinates and correspondingly changing the defining cubic polynomial h to
[TABLE]
with P3(y)=P3(Fy). In the (xy)-coordinates, let K denote the sectional curvature. By identifying ∂yk=∂yk=∂k for all 1≤k≤n (as the kth unit vector in Rn, not via the map F) we have
[TABLE]
∎
Another application of the first variation of the Pk’s as defined in Definition 3.7 is the study of homogeneous spaces that are
CCGPSR manifolds.
We will derive a sufficient condition for a connected GPSR manifold H⊂{h=1} to be a homogeneous space with respect to the action of G0h, that is the identity-component of the automorphism group Gh of h.
Proposition 3.12**.**
Let H be a maximal connected GPSR manifold in standard form.
Let δPk(y):Rn→Symk(Rn)∗ be as in equation (3.27) depending on dB0∈Lin(Rn,so(n)) (3.26), cf. Proposition 3.6. Then the connected component containing the neutral element of the automorphism group of h, that is G0h, acts transitively on H if and only if there exists a choice for dB0∈Lin(Rn,so(n)), such that δPk(y)≡0 for all 3≤k≤τ. Furthermore, each of the latter two equivalent statements imply that H is a CCGPSR manifold.
Proof.
It is clear that the action G0h×H→H is well defined. Assume that G0h acts transitively on the maximal GPSR manifold H⊂{h=1}. Then H is, in particular, a CCGPSR manifold. For p=(pxpy)∈H let M(p)∈G0h, such that M(p)⋅(10)=p. We will show that M(p) is necessarily of the form (3.6). We immediately see that M(p) is of the form
[TABLE]
for some vp∈Rn and W(p)∈Mat(n×n,R). We calculate
[TABLE]
Since by assumption h≡h∘M(p) it follows that
[TABLE]
and
[TABLE]
for all y∈Rn.
Suppose that W(p)∈GL(n). Then there exists y∈Rn∖{0}, such that W(p)y=0. Then by (3.41)
[TABLE]
This in particular shows that ⟨vp,y⟩=0. But then equation (3.40) cannot be fulfilled since ∂yhp(W(p)y)=∂yhp(0)=0 and ∂xh∣H>0 is true since H is a CCGPSR manifold, cf. proof of Proposition 3.1 equation (3.7). We deduce that W(p)∈GL(n). Hence, setting vp=W(p)Tvp in equation (3.40) implies that vp=−∂xh∂yhp. This shows that
[TABLE]
is of the form (3.6) as claimed. The action G0h×H→H might not be simply transitive, but near p=(10)∈H, that is on some open neighbourhood U⊂H of (10), we can choose a smooth branch of the possible maps W:U→GL(n) by the implicit function theorem. Then, using the diffeomorphism Φ:dom(H)→H (3.18), M∘Φ is locally on Φ−1(U) a valid choice for A as in equation (3.21) and d(W∘Φ)0 must fulfil the same equation as E in (3.26) in the proof of Proposition 3.6. We now use the equality
[TABLE]
for all z∈Φ−1(U) to conclude with the definition of the δPk’s (3.27) that there exists a linear map dB0∈Lin(Rn,so(n)), such that the corresponding functions δPk(y):Rn→Symk((Rn)∗) identically vanish for all y∈Rn and all 3≤k≤τ.
Now assume that there exists dB0∈Lin(Rn,so(n)), such that δPk(y)≡0 for all 3≤k≤τ. Consider the corresponding map A:Φ−1(V)→GL(n+1) (3.21) for any open neighbourhood V⊂H of the point (10)∈H so that A is defined, with
[TABLE]
cf. equations (3.26) and (3.24). Then δPk(y)≡0 for all 3≤k≤τ implies that for all v∈T0dom(H)≅Rn
[TABLE]
where dA0(v) denotes the gl(n+1)-valued 1-form dA at z=0 applied to v∈T0dom(H). With
[TABLE]
for 1≤i≤n, the set of matrices {a1,…,an} is linearly independent. Furthermore {a1,…,an}⊂T\mathbbm1Gh=T\mathbbm1G0h which follows from (3.42). Let μ:G0h→H, μ(a)=a⋅(10), denote the action of G0h on the point (10)∈H. Then dμ\mathbbm1(ai)=∂yi
for all 1≤i≤n. Hence, dμ\mathbbm1:T\mathbbm1G0h→T(10)H is surjective (recall that with h of the form (3.10), we view T(10)H as the vector subspace {(0v)v∈Rn}⊂Rn+1). This shows that there exists an open subset U⊂H, such that (10)∈U and U⊂G0h⋅(10).
Suppose that the orbit G0h⋅(10)⊂H is not open in H. Then the set H∩∂(G0h⋅(10))∩(G0h⋅(10)) is non-empty. Let q∈H∩∂(G0h⋅(10))∩(G0h⋅(10)) and let a(q)∈G0h, such that q=a(q)⋅(10). Since q is by assumption an element of ∂(G0h⋅(10)) and a(q) acts via linear transformations on Rn+1 restricted to H, there must exist p∈∂U, such that a(q)p=q, because otherwise q∈a(q)⋅∂U and q∈a(q)⋅U would imply that q∈∂(G0h⋅(10)). But we have by definition of G0h that G0h⊂GL(n+1) and, hence, (10)=a(q)−1q=p, this is a contradiction to p∈∂U. We conclude that the orbit G0h⋅(10)⊂H is open in H. Since H⊂Rn+1 is maximal and being a hyperbolic point of h is an open condition in Rn+1 it follows that H∩∂H=∅. This shows that the same also holds for the relative to H open orbit G0h⋅(10), i.e. that (G0h⋅(10))∩∂(G0h⋅(10))=∅ where the boundary of G0h⋅(10) is relative to Rn+1. This implies that G0h⋅(10) is an n-dimensional submanifold of Rn+1. Furthermore, (G0h⋅(10),gH∣G0h⋅(10)) is also by construction a homogeneous Riemannian manifold and, hence, in particular geodesically complete.
This implies that G0h⋅(10)⊂Rn+1 is closed, which can be seen the following way. Suppose that G0h⋅(10) is not closed in Rn+1 but geodesically complete with respect to the restriction of gH and let p0 be a point in the boundary ∂(G0h⋅(10)). For any other point p∈G0h⋅(10) consider a curve γ:[0,1)→G0h⋅(10) with γ(0)=p and t→1,t<1limγ(t)=p0. Since G0h⋅(10)⊂H⊂{h=1} and h:Rn+1→R, we conclude that 1=t→1,t<1limh(γ(t))=h(t→1,t<1limγ(t))=h(p0). Since gH=−τ1∂2h∣TH×TH it in particular follows from the fact that h(p0)=1 and that h is a homogeneous polynomial of homogeneity-degree τ that gH can be smoothly extended to p0∈G0h⋅(10).
This implies
[TABLE]
This is a contradiction to the geodesic completeness of (G0h⋅(10),gH∣G0h⋅(10)) (recall that by the Hopf-Rinow theorem a Riemannian manifold is geodesically complete if and only if all unbounded curves have infinite length).
By assumption, H⊂Rn+1 is maximal, and we have shown that G0h⋅(10)⊂Rn+1 is closed. We deduce that H=G0h⋅(10) and that the action of G0h on H is, in fact, transitive. In particular, H is a CCGPSR manifold.
∎
4 Construction of a compact convex generating set of the moduli set of CCPSR manifolds
In order to prove Theorem 1.1, broadly speaking we need to construct estimates for P3 and eigenvalues of its second derivative and study properties of dom(H), cf. Definition 3.3.
Lemma 4.1**.**
Let H be a CCPSR manifold in standard form.
Then
[TABLE]
Proof.
Consider f(t):=β(tz)=1−t2+t3P3(z), where β:Rn→R as in equation (3.20). Since dom(H) is precompact (Lemma 2.14), f must have at least one positive and one negative real root. We will determine the range for P3(z) such that this holds. The first and second derivative of f are
[TABLE]
Hence, f˙(t)=0 if and only if t=0 or t=3P3(z)2. We obtain f¨(0)=−2 and f¨(3P3(z)2)=2. This implies that f(t) has a local maximum at t=0 and a local minimum at t=3P3(z)2. If P3(z)=0, f(t)=0 if and only if t=±1, so in this case f(t) has exactly one positive and one negative real root. Now assume P3(z)>0. In that case, 3P3(z)2>0 and t→−∞limf(t)=−∞. Since f(0)=1, this implies that f(t) has at least one negative real root (one can show that it is the only negative real root by showing that f˙(t)>0 for all t<0 if P3(z)>0). We have seen that f(t) attains its unique local minimum at t=3P3(z)2. Furthermore f(0)=0, and t→∞limf(t)=∞. Hence, f(t) has a positive real root if and only if
[TABLE]
For P3(z)<0 we define f(t):=1−t2+t3(−P3(z)). Similarly as for P3(z) we then obtain −P3(z)≤332.
Summarising, we have shown that ∣P3(z)∣≤332.
∎
Note that the bounds (4.1) for P3(z), z∈{z∈Rn∣⟨z,z⟩=1}, are independent of the CCPSR manifold and of its dimension. We will later show that these bounds are in fact sharp and optimal in all dimensions, see Theorem 4.15. An immediate consequence of the calculations in Lemma 4.1 is the following.
Corollary 4.2**.**
Let h:Rn+1→R, h=x3−x⟨y,y⟩+P3(y), be a cubic homogeneous polynomial and let H denote the connected component of {h=1}⊂Rn+1 that contains the point (xy)=(10). Then the connected component of the set
[TABLE]
which contains the point (xy)=(10)
is precompact if and only if ∥z∥=1max∣P3(z)∣≤332.
Recall that we know from Lemma 2.14 that the connected component of the set {h>0}∩{(1z)∈Rn+1z∈Rn} that contains the point (xy)=(10) being pre-compact is a necessary condition for the connected component of {h=1} that contains (xy)=(10) to be a CCPSR manifold. Also note that if the connected component H⊂{h=1} that contains the point (xy)=(10) is a CCPSR manifold, then the connected component of the set {h>0}∩{(1z)∈Rn+1z∈Rn} that contains the point (xy)=(10), the set (R>0⋅H)∩{(1z)∈Rn+1z∈Rn}, and {1}×dom(H) coincide. One could ask if we can find similar bounds for CCGPSR manifolds of homogeneity-degree τ≥4. This is in general not true, see [Li, Lem. 7.9] for quartic CCGPSR manifolds, that is hyperbolic centro-affine hypersurfaces defined by quartic homogeneous polynomials.
Lemma 4.1 also means that we have determined positive and negative bounds for P3(z), z∈{z∈Rn∣⟨z,z⟩=1}, that guaranty that the corresponding hypersurface which is the connected component of {h=1} containing the point (10)∈Rn+1 is closed. However, it does at this point not give us information about hyperbolicity when we are studying some specific connected PSR manifold and want to know whether it is a CCPSR manifold or not. It will later turn out that this condition also shows hyperbolicity of all points contained in the connected component of {h=1} that contains the point (10)∈Rn+1, see Theorem 4.15.
Next, we will use Lemma 4.1 to determine upper and lower positive bounds for the norm of points in the boundary of dom(H)⊂Rn, that is ∂dom(H), corresponding to a CCPSR manifold H.
Lemma 4.3**.**
In the setting of Lemma 4.1, assume without loss of generality that P3(z)≥0. Let NP3(z) be the biggest negative real root of f(t) and PP3(z) be the smallest positive real root of f(t), where f(t) is associated to a CCPSR manifold H as in the previous lemma and ∣P3(z)∣≤332. Then
[TABLE]
Proof.
Let 0≤A<B≤332, and define
[TABLE]
fA(t) and fB(t) have a unique negative real root NA and NB, respectively. Furthermore, NA<NB. To see this we calculate f˙A(t)=−2t+3t2A and f˙B(t)=−2t+3t2B, from which it is immediate that f˙A(t)>0,f˙B(t)>0 for all t<0.
Since t→−∞limfA(t)=−∞, t→−∞limfB(t)=−∞, and fA(0)=fB(0)=1 this implies that NA and NB exist and are the unique negative real roots of fA(t), respectively fB(t). We further obtain
[TABLE]
Using f˙B∣t<0>0 this shows that
[TABLE]
We apply this result to NP3(z) and obtain −1=N0≤NP3(z)≤N332=−23.
The value of N332 can easily be found by checking that f332(t)=332(t+23)(t−3)2.
Now let PA and PB be the smallest positive root of fA(t) and fB(t), respectively. Then PA<PB. To see this, first note that the existence of PA and PB is ensured by the estimate (4.1) in Lemma 4.1. We obtain
[TABLE]
Since fA(0)=1 this shows that fA(t) has a positive real root that is smaller than PB, and in particular that
[TABLE]
Again, we apply this result to PP3(z) and obtain 1=P0≤PP3(z)≤P332=3.
∎
Lemma 4.3 implies the following result for the Euclidean norm of points in ∂dom(H).
Corollary 4.4**.**
For a CCPSR manifold H in standard form
and corresponding dom(H) as in Definition 3.3, the following holds true:
[TABLE]
where ∥⋅∥ denotes the norm with respect to the standard Euclidean scalar product ⟨⋅,⋅⟩ on Rn in the y-coordinates from equation (3.10).
Hence, with the notation Br(0)={z∈Rn∣⟨z,z⟩<r2}
for r>0, we have the inclusions B23(0)⊂dom(H)⊂B3(0) for all CCPSR manifolds H. In particular this is also independent of the point chosen in the process (see Proposition 3.1) of obtaining h in the form (3.10) for any given CCPSR manifold H⊂{h=1}. Note however that the inclusion B23(0)⊂dom(H) might not be compact in the sense that ∂B23(0)∩∂dom(H) might not be empty. If we choose any 0<R<23, BR(0) will always be compactly embedded via the inclusion in dom(H) since the inclusion BR(0)⊂B23(0) is a compact embedding.
Another consequence of Lemma 4.3 is the following characterisation of CCPSR manifolds that are singular at infinity, cf. Definition 2.15.
Lemma 4.5**.**
Let H be CCPSR manifold in standard form.
Then H is singular at infinity in the sense of Definition 2.15 if and only if ∥z∥=1max∣P3(z)∣=332.
Proof.
First note that with our assumptions for H and h, ∂(R>0⋅H)∖{0}=R>0⋅({1}×∂dom(H)). Since dhp is homogeneous of degree 2 in p, it thus suffices to show that there exists a z∈∂dom(H), such that dh(1z)=0 if and only if ∥z∥=1max∣P3(z)∣=332. In Lemma 3.5 we have shown that for z∈∂dom(H), dh(1z)=0 is equivalent to ∂x∂h((1z))=α(z)=0, which is by the Euler identity for homogeneous functions equivalent to dβz(z)=0. Hence, H is singular at infinity if and only if there exists a point z∈{∥z∥=1}, such that the 1-dimensional CCPSR manifold Hz defined by restricting h to the 2-dimensional linear subspace
[TABLE]
is singular at infinity.
More precisely, Hz is the connected component of {hz:=x3−xt2+t3P3(z)=1} that contains the point (xt)=(10), and we have {1}×dom(Hz)=E∩({1}×dom(H)). The corresponding function βz as in (3.20) for hz is given by
[TABLE]
Let t+ and t− denote the smallest positive root and the biggest negative root of βz(t), respectively. Then ∂dom(Hz)={t+z,t−z}. We have shown in Lemma 4.3 (with the notation βz(t)=f∣P3(z)∣(t)) that ∂tβz(t+)=0 or ∂tβz(t−)=0 if ∣P3(z)∣=332. It remains to show that ∣P3(z)∣<332 implies that ∂tβz(t) does not vanish at neither t+ nor t−. To do that, assume without loss of generality P3(z)≥0. For P3(z)<0 we can simply use the reflection t→−t and consider βz(−t). For P3(z)=0 it is easy to check that ∂tβz(t±)=∓2. Now assume P3(z)>0. We have
[TABLE]
hence ∂tβz(t−)>0 is always true and ∂tβz(t+)=0 if and only if t+=3P3(z)2. One quickly finds that βz(t+)=0 and P3(z)>0 if and only if P3(z)=332. This shows that ∂tβz vanishes at a point z∈dom(Hz)={t+z,t−z} (which is equivalent to Hz being singular at infinity) if and only if ∣P3(z)∣=332. Summarising, we have shown that there exists a point z∈∂dom(H), such that dβz(z)=0 if and only if there exists a point z∈∂dom(H), such that P3(∥z∥z)=332. In Lemma 4.3 we have shown that this is precisely the maximal possible value for ∣P3(z)∣ on {∥z∥=1} that does not exclude the property of H being closed in Rn+1. We conclude that ∥z∥=1max∣P3(z)∣=332 if and only if H is singular at infinity.
∎
Note that the set of CCPSR manifolds that are singular at infinity and of dimension n≥1 is not empty for all n≥1. This is one of the consequences of Theorem 1.1, but we can also use Proposition 3.12 and the above Lemma 4.5 to prove both the latter statement and a property of homogeneous CCPSR manifolds:
Proposition 4.6**.**
Homogeneous CCPSR manifolds are singular at infinity.
Proof.
Let H be a homogeneous CCPSR manifold and without loss of generality assume that H is in standard form. In Proposition 3.12 we have seen that H being homogeneous is equivalent to the existence of dB0∈Lin(Rn,so(n)) as in equation (3.23), such that δP3(y)≡0, cf. equation (3.28). Applying δP3(y) to the position vector field in Rn, we obtain
[TABLE]
for all y∈Rn, where dB0(y)y=k=1∑n(n−1)/2aky⟨ℓk,y⟩. Let now y∈{∥y∥=1} be a local positive maximum of P3∣{∥y∥=1}. This means that there is r>0, such that dP3∣y=r⟨y,dy⟩. Since dB0 has image in so(n), this implies using ∥y∥=1, (4.5), and 3P3(y,y,dy)=dP3∣y
[TABLE]
The above equation and y being a local positive maximum of P3∣{∥y∥=1} show that r=32. Using dP3∣y=32⟨y,dy⟩ and the Euler identity for homogeneous functions we find P3(y)=332. This is by Lemma 4.5 equivalent to H being singular at infinity.
∎
Note that, in theory, one could have also used the classification of homogeneous CCPSR manifolds given in [DV] for dim(H)≥3 and [CHM, CDL] for dim(H)=1 and dim(H)=2, respectively, to prove Proposition 4.6. One would then have to determine a standard form for the corresponding polynomials and could then use Lemma 4.5. This would, however, be most likely much more time-consuming.
Remark 4.7**.**
Note that the proof of Proposition 4.6 shows that every local positive maximum y∈{∥y∥=1} of P3∣{∥y∥=1} fulfils P3(y)=332 (and is thereby also a global maximum). It is an interesting open question whether this is enough to completely classify homogeneous CCPSR manifolds in the sense that this statement is equivalent to a CCPSR manifold being homogeneous.
We will now determine an estimate for the bilinear form P3(z,dz,dz) for all z∈dom(H). It will use the hyperbolicity property of the CCPSR manifold H, which we first need to reformulate.
Lemma 4.8**.**
Let h:Rn+1→R be a cubic homogeneous polynomial of the form (3.10), that is h=x3−x⟨y,y⟩+P3(y), and let H⊂{h=1} be the connected component of the level set {h=1}⊂Rn+1 that contains the point (xy)=(10).
Then H is a CCPSR manifold if and only if
[TABLE]
Proof.
Assumption that H is a CCPSR manifold. Then H fulfils the assumptions of this lemma and (R>0⋅H)∩{(1z)∈Rn+1z∈Rn} coincides with dom(H), cf. Definition 3.3. We will show that condition (4.6) follows from the hyperbolicity of each point in H. For each p∈H⊂Rn+1, the tangent space TpH viewed as a the hyperplane ker(dhp)⊂Rn+1 and the line Rp⊂Rn+1 are orthogonal with respect to the Lorenzian inner product −∂2hp. Recall that −∂2hp being Lorenzian precisely means that p is a hyperbolic point, see Definition 2.4. Since −∂2hp is homogeneous of degree 1 in p, it follows that the property that H consists only of hyperbolic points is equivalent to the statement that −∂2h(1z) is Lorenzian for all z∈dom(H).
Since −∂2h(10) is always Lorenzian if h is of the form (3.10), −∂2h(1z) being Lorenzian on dom(H) is equivalent to det(−∂2h(1z))<0 for all z∈dom(H). Consider
[TABLE]
Since (3\mathbbm1−9P3(z,⋅,⋅)+z⊗⟨z,⋅⟩)∣z=0=3\mathbbm1, it follows that det(−∂2h(1z))<0 for all z∈dom(H) is equivalent to 3⟨dz,dz⟩−9P3(z,dz,dz)+⟨z,dz⟩2>0 for all z∈dom(H).
For the other direction, the conditions that H is a connected component of {h=1} implies that it is closed as a subset of Rn+1. Furthermore, H is a hypersurface since dh does not vanish along H by the Euler identity for homogeneous functions. With the same argument as before for the homogeneity of −∂2hp in p and the same calculations as above, it follows that H consists only of hyperbolic points. H is thus a connected and also closed PSR manifold, and the set prRn((R>0⋅H)∩{(1z)∈Rn+1z∈Rn}) and dom(H) coincide.
∎
We will use the results from Corollary 4.4 and Lemma 4.8 to find upper and lower bounds of the eigenvalues of P3(z,dz,dz) (when viewed as a symmetric matrix) for z∈dom(H) that are valid for all CCPSR manifolds H (and thus also for non-connected closed PSR manifolds).
Proposition 4.9**.**
Let H be a CCPSR manifold in standard form.
Then
[TABLE]
This is equivalent to the statement that for all z∈dom(H), the eigenvalues λ∈R of the representation matrix of the symmetric bilinear form P3(z,dz,dz) induced by the z-coordinates fulfil −65<λ<32. Furthermore, the upper bound in (4.7) is sharp in the sense that for all n≥1 there exists a CCPSR manifold H and a point zˇ∈∂dom(H), such that the representation matrix of P3(zˇ,dz,dz) has one eigenvalue λ=32.
Proof.
We start with the upper bound in (4.7). Equation (4.6) in Lemma 4.8 and equation (4.4) in Corollary 4.4 imply for all z∈dom(H)
[TABLE]
Obtaining the alleged lower bound in equation (4.7) for P3(z,dz,dz) needs more work. An other, but worse, lower bound can be obtained the following way. For all zˇ∈dom(H) with ∥zˇ∥=23 (recall that B23(0)⊂dom(H) is always true, see Corollary 4.4), the biggest positive eigenvalue of the representation matrix of P3(zˇ,dz,dz) is bound from above by 32. Using that P3(z,dz,dz) is linear in z, we obtain that the smallest eigenvalue of the representation matrix of P3(−2zˇ,dz,dz) is bounded from below by −34. Since zˇ∈∂B23(0) was arbitrary, we obtain for all z∈∂B3(0) the estimate P3(z,dz,dz)≥−34⟨dz,dz⟩. Since for all CCPSR manifolds with the assumptions of this lemma dom(H)⊂B3(0), we can use the linearity of P3(z,dz,dz) in z again to conclude that for all z∈dom(H) we have the estimate P3(z,dz,dz)>−34⟨dz,dz⟩. This bound is worse than −65⟨dz,dz⟩, which we will derive now.
The estimate (4.8) shows that for all zˇ∈∂dom(H), every positive eigenvalue λ+ of the representation matrix of P3(zˇ,dz,dz) fulfils
[TABLE]
Fix zˇ∈∂dom(H)⊂R2 and let λ− be a negative eigenvalue of the representation matrix of P3(zˇ,dz,dz). The linearity of P3(z,dz,dz) in z implies that −λ− is a positive eigenvalue of the representation matrix of P3(−zˇ,dz,dz). However, −zˇ might not be an element of dom(H). In fact, −zˇ∈dom(H) if and only if ∥zˇ∥≤1, which holds if and only if P3(∥zˇ∥zˇ)∈[−332,0] (cf. Lemma 4.1 and see Figure 1).
For such a given zˇ∈∂dom(H) we want to find tˇ>0, such that tˇ(−zˇ)∈∂dom(H). When we have determined said tˇ, the linearity of P3(z,dz,dz) in z implies that tˇ(−λ−) is a positive eigenvalue of the representation matrix of P3(tˇ(−zˇ),dz,dz). Using the upper bound (4.9), we can thus estimate
[TABLE]
Our asserted lower bound −65 for λ− is now obtained via showing that the function F:[23,3]→R>0 defined in (4.10) is continuous and by determining its maximal value, where we recall that the elements in the closed interval [23,3] are precisely all possible values for ∥zˇ∥ when considering all possible n-dimensional CCPSR manifolds H (cf. Corollary 4.4). To find a closed formula for tˇ depending on ∥zˇ∥, consider the function f(t)=β(t∥zˇ∥zˇ)=1−t2+P3(∥zˇ∥zˇ)t3 (compare with equation (3.20)) and assume that f(∥zˇ∥)=0. By assumption, H is a CCPSR manifold, implying that dom(H)⊂Rn is precompact and, hence, f(t) must have at least one more negative real root in addition to its root t=∥zˇ∥>0. Hence, (t−∥zˇ∥)∣f(t) and we obtain with a,b∈R
[TABLE]
This implies that a=∥zˇ∥2−1 and b=∥zˇ∥1−∥zˇ∥31. Note that this determines P3(∥zˇ∥zˇ) depending on ∥zˇ∥, and as we would expect P3(∥zˇ∥zˇ)=332 if ∥zˇ∥=3, and P3(∥zˇ∥zˇ)=−332 if ∥zˇ∥=23 (see Lemma 4.1 and Corollary 4.4). We define
[TABLE]
In order to determine tˇ in dependence of ∥zˇ∥ we need to find the roots of f(t), for (at least) one of the roots coincides with tˇ(−∥zˇ∥). We will differentiate between the three cases ∥zˇ∥=1, ∥zˇ∥∈(1,3], and ∥zˇ∥∈[23,1). We will also use these results to show that F is continuous.
Case 1: ∥zˇ∥=1.
In that case f(t)=1−t2, so the roots of f(t) are t±=±1 and the root of f(t)=−1−t is t=−1. Hence, tˇ=1 and (4.10) thus yields the estimate −λ−≤94=F(1).
Case 2: ∥zˇ∥∈[23,3]∖{1}.
In this case,
[TABLE]
Note that the sign of ∥zˇ∥2−1 depends on whether ∥zˇ∥<1 or ∥zˇ∥>1. We will treat these cases separately.
Case 2.1: ∥zˇ∥∈(1,3].
In this case, the plot of f(t) is of the form as in Figure 2 (except when ∥zˇ∥=3, in which case f(t) has the unique positive double root 3).
Also ∥zˇ∥2−1>0 and, hence, t−=tˇ(−∥zˇ∥). We obtain
[TABLE]
and, hence,
[TABLE]
It is clear that F∣(1,3] is smooth. Using L’Hôspital’s rule for limits twice at ∥zˇ∥=1 yields
[TABLE]
which coincides with F(1) determined in Case 1. This means that F is continuous from the right at ∥zˇ∥=1. Next we will show that F∣(1,3] attains its maximum, namely at ∥zˇ∥=3. To prove that we show that ∂∥zˇ∥∂F(1,3)>0. The first derivative of F is given by
[TABLE]
and ∥zˇ∥−1>0 implies that in order to solve ∂∥zˇ∥∂F(∥zˇ∥)=0 with the restriction ∥zˇ∥∈(1,3) we only need to solve 8∥zˇ∥4−18∥zˇ∥2+9+4∥zˇ∥2−3=0. Using MAPLE or any other computer algebra system one finds that the latter equation has no solutions in (1,3). It thus suffices check the sign of ∂∥zˇ∥∂F(1,3) at one point in the interval, say 21+3, to determine its global sign. We calculate
[TABLE]
We conclude that
[TABLE]
Case 2.2: ∥zˇ∥∈[23,1).
This case works similarly to Case 2.1. Here, f(t) has the shape as in Figure 3 (except for the case ∥zˇ∥=23, where f(t) has the unique negative double root 23).
In this case, f(t) has, except if ∥zˇ∥=23, precisely two negative roots, of which we need to consider the bigger one. Since ∥zˇ∥2−1<0, we see that this is
[TABLE]
so that t+=tˇ(−∥zˇ∥) has the form
[TABLE]
We see that formally the function F for this case and F in Case 2.1 coincide, i.e. we have for F∣[23,1)
[TABLE]
and for the derivative ∂∥zˇ∥∂F(23,1)
[TABLE]
Proceeding analogously to Case 2.1 we will show that ∂∥zˇ∥∂F(23,1)>0. Note that the denominator of the formula for the first derivative of F has no zeros in (23,1), so we will not run into trouble with possibly singular values. Again, we use MAPLE to show that 8∥zˇ∥4−18∥zˇ∥2+9+4∥zˇ∥2−3=0 has no solutions in (23,1). Hence, the global sign of ∂∥zˇ∥∂F(23,1) coincides with the sign of
[TABLE]
and thus F[23,1) does not attain its maximum in its domain of definition, but at the limit ∥zˇ∥→1, assuming that limit exists. For the existence we need to check that F is continuous from the left at ∥zˇ∥=1. This is done in the same way we have shown continuity from the right, that is by applying L’Hôspital’s rule twice. As expected, we obtain
[TABLE]
Summarising, we have shown that F:[23,3]→R>0 is continuous and attains its maximum at ∥zˇ∥=3, F(3)=65. Since the negative eigenvalue λ− of the representation matrix of P3(zˇ,dz,dz) was arbitrary, we conclude with (4.10) that for all such negative eigenvalues λ− we have
[TABLE]
The point zˇ∈∂dom(H) was also arbitrary and, thus, using the linearity of P3(z,dz,dz) we obtain
[TABLE]
Note that our calculations also show that λ−=−65 can only possibly be a negative eigenvalue of the representation matrix of P3(zˇ,dz,dz) at a point zˇ∈∂dom(H) with norm ∥zˇ∥=3.
We want to stress again at this point that the obtained lower and upper bounds for P3(z,dz,dz) hold for all CCPSR manifolds H⊂{h=1} of dimension n≥1 with h of the form (3.10) and (10)∈H.
It remains to show that the upper bound in (4.7) is sharp in the stated sense.
To do so, we will give an example of a CCPSR manifold of dimension n for each n∈N. For any n∈N, let (xy)=(x,y1,…,yn)T denote linear coordinates of Rn+1 as usual and consider the cubic polynomial
[TABLE]
and the corresponding centro-affine hypersurface H⊂{h=1}, which is the connected component of {h=1} that contains the point (xy)=(10). Then H is a CCPSR manifold of dimension n. We will not prove this here,
since it follows from Theorem 1.1. Note that while this proposition is used in the later proof of the latter theorem, the sharpness of the upper bound is not required therein.
To show that the upper bound is in fact sharp in the stated sense, consider the point zˇ+=(0,…,0,3)T∈∂dom(H).
We obtain P3(zˇ+,dz,dz)=32dzn2 and the corresponding symmetric matrix has precisely the eigenvalues λ1=0 with eigenspace-dimension n−1, and λ2=32 with eigenspace-dimension 1.
This proves our claim.
∎*
*
Next, we will study the boundary behaviour of the centro-affine fundamental form of CCGPSR manifolds. For an explanation why this term is used, see the discussion under [CNS, Thm. 1.18] and the related chapter in Melrose’s book [M, Ch. 8].
Definition 4.10**.**
Let H⊂{h=1} be a CCGPSR manifold of dimension n≥1 and let U=R>0⋅H be the corresponding convex cone (cf. Proposition 2.13). Then H has regular boundary behaviour if
(i)
dhp=0* for all p∈∂U∖{0}, i.e. H is not singular at infinity in the sense of Definition 2.15,*
2. (ii)
−∂2hT(∂U∖{0})×T(∂U∖{0})≥0* and dimker(−∂2hT(∂U∖{0})×T(∂U∖{0}))=1 for all p∈∂U∖{0}.*
Note that Definition 4.10 is equivalent to [CNS, Def. 1.17] restricted to CCGPSR manifolds. We also want to stress that Definition 4.10 is independent of the chosen linear coordinates of the ambient space Rn+1.
Remark 4.11**.**
With the functions α and β as in (3.19) and (3.20),
Lemma 3.5 shows that the conditions (i) and (ii) in Definition 4.10 are equivalent to
(i)
dβz(z)=0 for all z∈∂dom(H),
2. (ii)
−∂2βz>0 for all z∈∂dom(H),
respectively.
It turns out that for CCPSR manifolds the condition Def. 4.10(i) always implies Def. 4.10(ii):
Theorem 4.12**.**
A CCPSR manifolds of dimension n≥1 is not singular at infinity in the sense of Definition 2.15 if and only if it has regular boundary behaviour as defined in Definition 4.10.
Proof.
A CCPSR manifold H that has regular boundary behaviour is by definition not singular at infinity. For the other direction, consider first n=1. Then Def. 4.10(ii) is trivially satisfied.
To prove the statement of this theorem for n≥2, it suffices to prove it for n=2. To see this, consider any CCPSR manifold H of dimension n>2 and assume that Def. 4.10(i) holds for H. Assume without loss of generality that H is in standard form.
Considering Remark 4.11, Def. 4.10(ii) holds true if and only if Rem. 4.11(ii) holds true. To show the latter we need to show that −∂2βz(v,v)>0 for all z∈∂dom(H) and all 0=v∈Tz∂dom(H)⊂Rn. Observe that for any 2-dimensional linear subspace E=span{w1,w2}⊂Rn, where w1 and w2 are chosen such that they are orthonormal with respect to ⟨⋅,⋅⟩, the restricted polynomial
[TABLE]
defines a 2-dimensional CCPSR manifold HE⊂{hE=1}⊂R3 as the connected component containing the point (xt1t2)=(100). Furthermore,
[TABLE]
is an embedding. Note that the explicit formula for hE in general depends on the choice of basis for E. Hence, if we want to show that −∂2βz(v,v)>0 for some fixed z∈∂dom(H) and 0=v∈Tz(∂dom(H)), it suffices to show Rem. 4.11(ii) for HE and hE, respectively βE((t1t2))=hE((1t1t2)), with E=span{z,v}555Strictly speaking, at this point we need to choose two orthonormal vectors w1,w2, such that span{z,v}=span{w1,w2}. where we view v as an element of Rn. Hence, proving the statement of this theorem for all 2-dimensional CCPSR manifolds will also prove it for these of higher dimension. Since the conditions in Definition 4.10 are independent of the linear coordinates chosen for the ambient space Rn+1, we can reduce our studies to the classification of 2-dimensional CCPSR manifolds up to equivalence given in [CDL, Thm. 1]666At the time the article [CDL] was written and published, it was still an open problem to show that a PSR manifold is closed if and only if it is geodesically complete, which has first been proven in [CNS, Thm. 2.5]., see Theorem 2.17. We will do a case-by-case check for the surfaces a)–e) and the one-parameter family of surfaces f) in Theorem 2.17. For the cases a)–e) we will study the P3-part the calculated standard form h=x3−x(y2+z2)+P3((yz)) (3.10) of each cubic h corresponding to a CCPSR surface H⊂{h=1} obtained in Example 3.2 with the property that H is equivalent to the connected component of {h=1} that contains the point (xyz)=(100). We can then use Lemma 4.5, which says that the value of ∥(yz)∥=1maxP3((yz))∈[0,332] determines whether H is singular at infinity or not. In the cases where H is not singular at infinity, that is fulfils Def. 4.10(i), we need to show that it also fulfils Def. 4.10(ii).
For the one-parameter family f) we will use another method and explain why in this case the form (3.10) is not the best choice to work with in order to prove our claim.
a) H={h=xyz=1,x>0,y>0}.
Equation (3.11) implies that P3((yz))=−332y3+32yz2. Since H is a CCPSR surface and P3((−10))=332, Lemma 4.5 implies that H is singular at infinity.
b) H={h=x(xy−z2)=1,x>0}.
By equation (3.12), P3((yz))=332y3+31yz2 with P3((10))=332. Hence, H is singular at infinity.
c) H={h=x(yz+x2)=1,x<0,y>0}.
This case is a little more complicated in comparison with a) and b). Equation (3.13) implies that P3((yz))=1522y2z+1515142z3.
We now need to determine ∥(yz)∥=1maxP3((yz)). We find for v=(223225), ∥v∥=1, that P3(v)=332. Hence, H being closed and connected implies that ∥(yz)∥=1maxP3((yz))=332. This shows that H is singular at infinity. Note that v can be found without the help of a computer algebra system like MAPLE by considering the equation dP3∣(yz)=r⟨(yz),⋅⟩, r>0, which is not difficult to solve in this case since P3((yz)) is reducible.
d) H={h=z(x2+y2−z2)=1,z<0}.
From equation (3.14) we obtain that in this case P3((yz))≡0. Hence, ∥(yz)∥=1maxP3((yz))=0 and H is thus not singular at infinity. It is immediate that dom(H)={(yz)<1} and that for the corresponding function β(y,z)=1−y2−z2 as in (3.20) we have dβ=−2ydy−2zdz. Hence, dβ vanishes at no point in ∂dom(H), so Lemma 3.5 implies that H, and thus also H, fulfils Def. 4.10(i). Furthermore
From equation (3.15) we know that P3((yz))=332y3−231yz2. Hence, P3((10))=332, which shows that H is singular at infinity.
f)777For this one-parameter family of CCPSR surfaces which are each contained in the level set of the respective Weierstaß cubic with positive discriminant h, the method used for a)–e) has proven itself to be unsuitable. This is because the formulas for the corresponding function β as in (3.20) and the derivatives corresponding to h when brought to the form (3.10) might not depend on b∈(−1,1) in a complicated way, but studying the system of equations v∈kerdβ, v∈ker∂2β, β=0, turned out to be quite difficult. We will thus consider Definition 4.10 and not the equivalent conditions in Remark 4.11 to prove our claim for this one-parameter family. Hb={h=y2z−4x3+3xz2+bz3=1,z<0,2x>z}, b∈(−1,1).
For all b∈(−1,1), the projective curve C:={h=y2z−4x3+3xz2+bz3=0}⊂RP2 has no singularities, cf. [CDL, Prop. 3], which means that dhp=0 for all p∈{h=0}∖{0}⊂R3. Hence, each Hb, b∈(−1,1), is not singular at infinity in the sense of Definition 2.15 and, hence, fulfils condition Def. 4.10(i). Note that Hb not being singular at infinity for all b∈(−1,1) also follows easily from equation (3.16) in Example 3.2. We need to show that each Hb, b∈(−1,1), also fulfils Def. 4.10(ii). In order to prove this, we need to determine ∂(R>0⋅Hb)⊂{h=0,z≤0,2x≥z}⊂R3 for each b∈(−1,1). Observe that {h=0,z≤0,2x≥z}∩{z=0}={x=0,z=0}.
Hence, the line {x=0,z=0} is contained in {h=0,z≤0,2x≥z}, but R>0⋅Hb being a convex cone which has the property described in Lemma 2.14 shows that {x=0,z=0}∩∂(R>0⋅Hb)={(000)}. For z<0 we will determine the intersection {z=−1}∩∂(R>0⋅Hb), which can then be used with the homogeneity of h to obtain the whole set ∂(R>0⋅Hb). We find
[TABLE]
where ρb((xy))=h((xy−1)). We consider ρb to be defined for all b∈R, not just for b∈(−1,1). Let
[TABLE]
and observe that Hb not being singular at infinity implies that the tangent space T∂(R>0⋅Hb) fulfils
[TABLE]
Furthermore, the 1-dimensional linear subspaces R⋅p and kerdhp∩V of Tp∂(R>0⋅Hb) are orthogonal with respect to the positive-semidefinite bilinear form −∂2hp, which follows from −∂2hp(p,⋅)=−2dhp(⋅). Also note that kerdhp∩V is always 1-dimensional since the position vector p=0 is always an element of kerdhp for all p∈∂(R>0⋅Hb). Thus, in order to prove that Def. 4.10(ii) is fulfilled for each Hb, b∈(−1,1), it suffices to show that −∂2h∣(kerdhp∩V)×(kerdhp∩V)>0. We obtain
[TABLE]
and
[TABLE]
Since Hb is not singular at infinity, it follows that at each point p=(xyz)∈{z=−1}∩∂(R>0⋅Hb), kerdhp∩V is given by
[TABLE]
Hence, −∂2h∣(kerdhp∩V)×(kerdhp∩V)>0 if and only if
[TABLE]
for all p=(xyz)∈{h=0,z=−1}∩∂(R>0⋅Hb). We will first check the above inequality (4.12) for y=0. In that case, (4.12) can only be false if x=±21. Then with ρb defined as in (4.11) we obtain
[TABLE]
This is however a contradiction to b∈(−1,1) and, hence, (4.12) holds at all points in {z=−1,y=0}∩∂(R>0⋅Hb). Now let y=0. We see that then (4.12) is true for all x≥0, independent of b∈(−1,1). It thus remains to check the inequality (4.12) for points in {z=−1,x<0}∩∂(R>0⋅Hb). Note that the latter set might be empty, in fact one can show that it is empty if and only if 0<b<1, but we will not need this information for our proof. Observe that for all b1,b2∈R with b1<b2,
[TABLE]
for all (xy)∈R2. Hence, ρb1∣{ρb2=0}>0, which in particular implies that
[TABLE]
for all b∈(−1,1). With the fact that (210−1)∈{z=−1}∩(R>0⋅Hb), cf. Example 3.2, and ρ−1((210))=2>0 it follows that {z=−1}∩(R>0⋅Hb) is a subset of the connected component of {ρ−1>0}×{−1}⊂R3 that contains the point (210−1), see Figure 4.
Further observe that ρb((−21y))=−y2−1−b<0 for all y∈R and b∈(−1,1), and that Hb⊂{z<0,2x>z} implies that {z=−1}∩∂(R>0⋅Hb) is contained in {z=−1,x>−21} for all b∈(−1,1). In particular there exists no b∈(−1,1), such that the x-coordinate of an element in {z=−1}∩∂(R>0⋅Hb) has the value −21. Hence, (4.12) and (4.13) imply that in order to prove that Hb fulfils Def. 4.10(ii) it suffices to show
[TABLE]
since (16xy2+(43x2−3)2){x>−21,y=0}>0, see also Figure 5.
We insert ρ−1=0, which is equivalent to y2=−4x3+3x+1, into 16xy2+(43x2−3)2=0 and obtain
[TABLE]
One can now use a computer algebra system like MAPLE and find that F(x)=0 and −21≤x<0 if and only if x=−21. This proves (4.14) and, hence, shows that each Hb, b∈(−1,1), fulfils Def. 4.10(ii).
This finishes the proof of Theorem 4.12.
∎*
*
Lemma 4.5 and Theorem 4.12 imply the following.
Corollary 4.13**.**
A CCPSR manifold in standard form H⊂{h=x3−x⟨y,y⟩+P3(y)=1}
has regular boundary behaviour if and only if ∥z∥=1maxP3(z)<332.
Next, we will prove a property of CCPSR manifolds in standard form which already implies that the moduli set of CCPSR manifolds is generated by a path-connected subset of Sym3(Rn)∗.
Proposition 4.14**.**
Let H⊂{h=x3−x⟨y,y⟩+P3(y)=1} be a CCPSR manifold in standard form.
Then for all s∈[0,1], the connected component Hs⊂{hs:=x3−x⟨y,y⟩+sP3(y)=1} that contains the point (xy)=(10) is a CCPSR manifold.
Proof.
For all s∈[0,1],
[TABLE]
Hence Corollary 4.2 shows that for each corresponding Hs, which is by definition closed as a subset of Rn+1, the necessary condition for Hs to be a CCPSR manifold, namely that the set (R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn}⊂Rn+1 is precompact, is satisfied. For s=1, H1 and H coincide. For s=0, (4.6) in Lemma 4.8 immediately shows that H0 is a CCPSR manifold. Now consider s∈(0,1) and let (1z)∈(R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn} be arbitrary. For z=0, (4.6) in Lemma 4.3 is always true. For z=0, we will differentiate between the cases P3(z)≥0 and P3(z)<0. In the first case, that is P3(z)≥0, the estimate (4.3) in Lemma 4.1 for fsP3(∥z∥z)(t)=hs((1t∥z∥z)) (note: B=P3(∥z∥z) and A=sP3(∥z∥z)) show that z∈dom(H) for all s∈(0,1). Hence, using the hyperbolicity of H we estimate
[TABLE]
This shows that all points in (R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn} with P3(z)≥0 satisfy (4.6) in Lemma 4.8 for all s∈(0,1).
Next, consider the case P3(z)<0. This case is a bit more complicated, since the estimate (4.2)888With corresponding values B=−P3(∥z∥z) and A=−sP3(∥z∥z). in Lemma 4.3 for f−sP3(∥z∥z)(t)=hs((1−t∥z∥z)) shows that for all s∈(0,1) there exist points in (R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn} that are not contained in the set
Consider for z∈R>0⋅z, such that z∈∂dom(H), and for t∈[0,1] the function r:[0,1]→[1,∞) implicitly defined by
[TABLE]
The condition that r(t) is a positive function and the uniqueness of the positive real root of r↦F(r,t) for all t∈[0,1] show that F(r,t)=0 indeed defines r(t) in a unique way, and furthermore that r(t) is smooth for t∈(0,1) and continuous for t∈[0,1] (note: P3(z)<0).
The map
[TABLE]
is thus a diffeomorphism for all s∈[0,1]. Furthermore, Ψ can be continuously extended to be defined on {1}×((R>0⋅z)∩dom(H)) for all s∈[0,1], with the property that
[TABLE]
We obtain for the first t-derivative of r=r(t) for all t∈(0,1)
[TABLE]
Since P3(z)<0 and t∈(0,1), this in particular shows that r˙(t)>0 for all t∈(0,1). If the considered point (1z)∈(R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn} is also an element of {1}×dom(H), then we can use estimate (4.15) for all s∈(0,1). For (1z)∈(R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn}∖({1}×dom(H)), we want to show that (4.6) holds for all s∈(0,1), i.e. that 3⟨dz,dz⟩−9sP3(z,dz,dz)+⟨z,dz⟩2>0 for all s∈(0,1). Substituting s=1−t and z=prRnΨ(z)=r(t)z with z∈dom(H), the latter is equivalent to
[TABLE]
Since H is a CCPSR manifold by assumption, we already know that
[TABLE]
for all z∈dom(H), cf. Lemma 4.8. Since r2(t)>1 for all t∈(0,1), proving
[TABLE]
for all t∈(0,1) and all z∈dom(H) will in particular prove (4.17). Since the estimate 3⟨dz,dz⟩+⟨z,dz⟩2>0 holds true for all z∈Rn, it suffices to show that (1−t)r(t)≤1 for all t∈(0,1). The function (1−t)r(t) is non-negative and continuous on [0,1], and positive and smooth on (0,1). For t=0, (1−t)r(t)∣t=0=r(0)=1. Using (4.16) yields
[TABLE]
for all t∈(0,1).
Hence, 0≤(1−t)r(t)≤1 for all t∈[0,1]. This thus proves (4.17).
Summarising, we have shown that 3⟨dz,dz⟩−9sP3(z,dz,dz)+⟨z,dz⟩2>0 for all (1z)∈(R>0⋅Hs)∩{(1z)∈Rn+1z∈Rn} for all s∈[0,1], and thus have proven using Lemma 4.8 that Hs is a CCPSR manifold for all s∈[0,1].
∎
An immediate consequence of Proposition 4.14 is that we can always find a continuous curve connecting two CCPSR manifolds of the same positive dimension that consists pointwise of CCPSR manifolds. However, we will prove a stronger result in the following Theorem 4.15, from which it will in particular follow how such an aforementioned curve can look like (see Corollary 4.17).
Theorem 4.15**.**
Let n∈N and h:Rn+1→R be a cubic homogeneous polynomial of the form (3.10), that is h=x3−x⟨y,y⟩+P3(y). Then the connected component H of the level set {h=1}⊂Rn+1 that contains the point (xy)=(10) is a CCPSR manifold if and only if ∥z∥=1maxP3(z)≤332.
Proof.
Firstly note that P3:Rn→R being a cubic homogeneous polynomial and, hence, an odd function implies that ∥z∥=1maxP3(z)=∥z∥=1max∣P3(z)∣. Assume that H is a CCPSR manifold. Then Lemma 4.1 shows that ∥z∥=1maxP3(z)≤332.
Now assume that ∥z∥=1maxP3(z)≤332. Lemma 4.1 only shows that this is a necessary requirement for H to be a CCPSR manifold.
In order to show that it is also a sufficient condition, we have to show that
[TABLE]
for all (1z)∈(R>0⋅H)∩{(1z)∈Rn+1z∈Rn} and all v∈Rn∖{0}, cf. Lemma 4.8. For z=0, (4.18) is always true.
For z=0 and v=rz, r=0, (4.18) reads r2(3⟨z,z⟩−9P3(z)+⟨z,z⟩2)>0. Suppose that there exists a point (1z)∈(R>0⋅H)∩{(1z)∈Rn+1z∈Rn}∖{(10)}, such that 3⟨z,z⟩−9P3(z)+⟨z,z⟩2=0. Observe that ∥z∥=1maxP3(z)≤332 implies
[TABLE]
The map ∥z∥↦3−23∥z∥+∥z∥2 is non-negative and its only zero is at ∥z∥=3. Hence, for ∥z∥>0, ∥z∥2(3−23∥z∥+∥z∥2)=0 if and only if ∥z∥=3. Since by assumption (1z)∈R>0⋅H∩{(1z)∈Rn+1z∈Rn}∖{(10)}, we have h((1z))=1−⟨z,z⟩+P3(z)>0. But with ∥z∥=3,
[TABLE]
which is a contradiction. We conclude that whenever z=0 and v=0 are linearly dependent, the estimate (4.18) holds. Note that this already finishes the proof for n=1.
Now assume that dim(H)≥2 and let (1z)∈(R>0⋅H)∩{(1z)∈Rn+1z∈Rn}∖{(10)} be arbitrary. Let v∈Rn∖{0}, such that z and v are linearly independent. In order to show (4.18), choose an orthonormal basis {e1,e2} of span{z,v}⊂Rn with respect to ⟨⋅,⋅⟩ and consider the cubic homogeneous polynomial hˇ:R3→R given by
[TABLE]
Let Hˇ be the connected component of the level set {hˇ(z,v)=1}⊂R3 that contains the point (100)∈R3 and observe that
[TABLE]
via the linear map (xab)↦(xae1+be2). Hence, if we prove that the inequality (4.6) in Lemma 4.8 holds for all cubic homogeneous polynomials hˇ(z,v) of the form (4.19) with corresponding set (4.20), we will also have proven (4.6) in Lemma 4.8 for our considered h with corresponding set (R>0⋅H)∩{(1z)∈Rn+1z∈Rn} (recall that for z and v linearly dependent, (4.18) has already been shown to hold true). Furthermore note that
[TABLE]
We thus see that it suffices to prove the statement of this theorem for all considered manifolds H with the additional restriction dim(H)=2 in order to conclude that it holds true for all H with dim(H)≥2. In the following, we will use the notation used in [CDL] and consider R3 with linear coordinates (xyz),
[TABLE]
such that
[TABLE]
As before, we consider the centro-affine surface H which is the connected component of the level set {h=1}⊂R3 that contains the point (xyz)=(100), and we want to show that H is a CCPSR surface (which is equivalent to the condition (4.6) in Lemma 4.8). For P3≡0, the condition (4.6) in Lemma 4.8 is immediately seen to be true. For P3≡0, Proposition 4.14 implies that it suffices to prove that H is a CCPSR surface if ∥(yz)∥=1maxP3((yz))=332, since for all non-vanishing cubic homogeneous polynomials P3:R2→R with ∥(yz)∥=1maxP3((yz))<332 we can always choose a positive real number r>0, such that ∥(yz)∥=1maxrP3((yz))=332. Consequently assume that ∥(yz)∥=1maxP3((yz))=332. We can, after a possible orthogonal transformation of the (yz)-coordinates (which in particular does not change the form (3.10) of h),
assume that P3∣{∥(yz)∥=1} attains its maximum at (yz)=(10), so that P3 is of the form
[TABLE]
We immediately see that ℓ∈R needs to fulfil ∣ℓ∣≤332. Furthermore, we can without loss of generality assume that ℓ≥0, which can be achieved via z↦−z if necessary.
Now we will show that for all ℓ∈[0,332], ∥(yz)∥=1maxP3((yz))=332 implies
[TABLE]
It will become clear how to use this information in the step thereafter.
First assume ℓ=0, so that P3((yz))=332y3+kyz2. We want to determine the positive extremal values and corresponding critical points of P3 when restricted to the set {(yz)=1} aside from 332, respectively (yz)=(10). Suppose that there exists k>31 or k<−32, such that ∥(yz)∥=1maxP3((yz))=332. In order to find the extremal values of P3 on {(yz)=1} we need to solve dP3∣(yz)=r⟨(yz),(dydz)⟩, r∈R, that is
[TABLE]
We already know that (yz)=(10) is an extremal point with P3>0, so we assume now that z=0. Then by (4.22) r=2ky, which implies
[TABLE]
Note that
[TABLE]
so (4.23) will always have non-trivial solutions. For k>31 or k<−32 consider the two points
[TABLE]
One quickly checks that ∥η±∥=1
and that η± both solve equation (4.23). We obtain
[TABLE]
and
[TABLE]
Furthermore,
[TABLE]
and we see that
[TABLE]
This shows that for ℓ=0 there exists no k∈R∖[−32,31], such that ∥(yz)∥=1maxP3((yz))=332.
It remains to consider the case ℓ∈(0,332]. For P3((yz))=332y3+kyz2+ℓz3 we get
[TABLE]
(note that ∥η±∥=1 independently of the chosen ℓ). Since
[TABLE]
it follows that
[TABLE]
for all k∈R∖[−32,31]. With
[TABLE]
and
[TABLE]
we can now conclude that for all ℓ>0, i.e. in particular for all ℓ∈(0,332], we have P3(η+)>332 and P3(η−)<−332.
Summarising, we have shown that for all k∈R∖[−32,31] and all ℓ∈[0,332]
[TABLE]
which in particular implies that for all ℓ∈[0,332], ∥(yz)∥=1maxP3((yz))=332 implies k∈[−32,31] as claimed in (4.21).
Next, we will deal with the cases with
[TABLE]
Equations (4.24) and (4.26) (for the lower limit k=−32) imply that for k=−32 and all ℓ∈(0,332]
[TABLE]
Hence, for k=−32, ℓ=0 is the only allowed value for ℓ∈[0,332] such that ∥(yz)∥=1maxP3((yz))=332.
The corresponding connected component H of {h=1} is equivalent to the CCPSR surface a) in Theorem 2.17, cf. equation (3.11) after a sign-flip in y and, hence, in particular a CCPSR manifold. The case k=31 is a little more complicated since then η±=(10), for which in particular ∂ℓP3(η±) vanishes, see (4.25) and (4.26). Instead of η± consider for ℓ≥0 the point
[TABLE]
One can check that dP3∣p∈R⟨p,⋅⟩ and
[TABLE]
For ℓ=0 we have P3(p)=332 and since ∂ℓ(P3(p))>0 for all ℓ>0 we deduce that
[TABLE]
This proves that for k=31, ℓ=0 is the only value allowed for ℓ∈[0,332]. For k=31, ℓ=0, the connected component H of {h=1} is equivalent to the CCPSR surface b) in Theorem 2.17 which follows from equation (3.12). Hence, H is a CCPSR manifold.
Now, as stated before, we will use (4.21). Considering (4.6) in Lemma 4.8 for points in the set
[TABLE]
yields
[TABLE]
With (4.21), that is k∈[−32,31], and y∈(−23,3)
we deduce
[TABLE]
This means that the line segment {(1y0)∈R3y∈(−23,3)}⊂R3 consists only of hyperbolic points of h, independently of the choice of k∈[−32,31], and the same statement is of course also true if we project it
to H via point-wise multiplication with 3h((1,y,0)T)1. Since being a hyperbolic point of h is an open condition in R3, we are in the setting of Proposition 3.1 and can transform h with linear transformations of the form (3.6) along that set999Note that this subset of H is connected and contains the point (1,0,0)T. Furthermore ∂xh=3x2−y, which is positive at all points (1,y,0)T, y∈(−23,3). Hence, we can in fact transform h along these points via transformations of the form (3.6)., that is along
[TABLE]
In order not to confuse coordinates with parametrisation of said subset of H, we replace y in the above set with the parameter T∈(−23,3). We start with E=\mathbbm1 in (3.6) and assign for T∈(−23,3)
[TABLE]
We obtain
[TABLE]
Note that 1−T2+332T3>0 and 1−kT>0 for all T∈(−23,3) and all k∈[−32,31], which is in accordance with
the positivity of the bilinear form in equation (3.5).
We have already shown that for k∈{−32,31}, ∥(yz)∥=1maxP3((yz))=332 implies ℓ=0 and that the corresponding surfaces H are indeed CCPSR manifolds. We will from here on assume that k∈(−32,31). Before bringing h in (4.28) to the standard form (3.10) we will check that we can always solve k−3T+32T=0 (the left hand side of which can be viewed as the “transformed k”, up to scale) for k∈(−32,31). We obtain
[TABLE]
We have to check that for all k∈(−32,31), T(k)∈(−23,3). For the limit points k∈{−32,31} we have
[TABLE]
and
[TABLE]
for all k∈(−32,31). Hence,
[TABLE]
as required. Considering (4.28), we rescale y and z with
[TABLE]
and set T=T(k) to obtain that h is equivalent to
[TABLE]
The next question one has to ask is if k∈(−32,31) and ∥(yz)∥=1maxP3((yz))=332 (for the P3-term in h, i.e. P3((yz))=332y3+kyz2+ℓz3) imply
[TABLE]
which is a necessary requirement for
[TABLE]
and thus also a necessary requirement that the transformed cubic in (4.30) needs to fulfil so that the corresponding connected component of its level set {h(A(T)⋅(1E(T))⋅(xyz))=1} which contains the point (xyz)=(100) can be a CCPSR manifold, cf. Corollary 4.2. Instead of attempting to calculate the supremum of ℓ(1−kT(k))231−T(k)2+332T(k)3 with conditions k∈(−32,31) and ∥(yz)∥=1maxP3((yz))=332 directly, we will choose another way to prove that (4.31) does, in fact, hold true.
For k=0, h is of the form h=x3−x(y2+z2)+332y3+ℓz3. Consider for T∈(−23,3) arbitrary, A(T) and E(T) as in (4.27) and (4.29), respectively,
[TABLE]
For the following calculations, we define
[TABLE]
We will show that
[TABLE]
holds true. To do so we will for T∈(−23,3) and ℓ=332 study a critical point of
[TABLE]
on the set {(yz)=1}, namely the point
[TABLE]
Note that ζ is well-defined for all T∈(−23,3), and it is indeed a critical point of P(3,332,T)((yz)). Using the factorisation 1−T2+332T3=332(T−3)2(T+23) and T−3<0 for all T∈(−23,3), we find
[TABLE]
The corresponding critical value is given by P(3,332,T)(ζ)=332,
independent of T∈(−23,3). Note that dz(ζ)>0 for all T∈(−23,3) and consider the derivative
[TABLE]
Hence, ∂ℓ(P(3,ℓ,T)(yz))>0 for all T∈(−23,3) and all z>0, in particular for z=dz(ζ). We conclude that (4.33) holds true.
We can now use (4.33) to show that (4.31) holds true for all k∈(−32,31). For ℓ=0 equation (4.31) is automatically true independently of the chosen k∈(−32,31). Suppose that there exist k∈(−32,31) and ℓ∈(0,332], with corresponding polynomial h=x3−x(y2+z2)+332y3+kyz2+ℓz3,
fulfilling
[TABLE]
such that for T=T(k)=2−3k3k
[TABLE]
which precisely means that (4.31) does not hold true for the chosen k, ℓ. Combining (4.30) and (4.32), one obtains that h is equivalent to
[TABLE]
for all T∈(−23,3). Furthermore, (4.33) implies that for all T∈(−23,3):
[TABLE]
The above estimate (4.35) must thus in particular hold for T=−23k=:T(k) (note that T(k)∈(−23,3) for all k∈(−32,31)). But
[TABLE]
which implies that (4.35) for T=T(k) is a contradiction to the assumption (4.34). We conclude that (4.31) holds true.
In order to complete the proof of this theorem it thus suffices to show that for all ℓ∈[0,332] and corresponding polynomial hℓ:=x3−x(y2+z2)+332y3+ℓz3, the connected component Hℓ⊂{hℓ=1} that contains the point (xyz)=(100) is a CCPSR manifold. Define the P3-part of hℓ as P(3,ℓ)((yz)):=332y3+ℓz3. One can easily check that
[TABLE]
which shows that ∥(yz)∥=1maxP(3,ℓ)((yz))=332 as required. We use the linear transformation
[TABLE]
and transform hℓ to
[TABLE]
In the new coordinates, H˘ℓ:=B−1(Hℓ)⊂{h˘=1} is given by
[TABLE]
This follows easily from B⋅(1−30)=(100)∈Hℓ and that x→∞ for all sequences in {y<0,y2>z2}={y≤0,y2≥z2} that converge to a point in ∂{y<0,y2>z2}={y≥0,y2=z2}. The latter follows from 332+ℓ≤334<1 for all ℓ∈[0,332]. We know that (1−30)=B−1⋅(100)∈H˘ℓ is always a hyperbolic point of h˘ℓ for all ℓ∈[0,332]. Hence, in order to show that H˘ℓ consists only of hyperbolic points of h˘ℓ, it suffices to show that
[TABLE]
for all (yz)∈{y<0,y2>z2}. The prefactor y2−z21 is always positive if y2>z2, and the term −y2+z2 is always negative. Hence, it suffices to show Rℓ(y,z)≤0 for all (yz)∈{y<0,y2>z2}. We calculate
[TABLE]
which implies that Rℓ(y,z) vanishes on ∂{y<0,y2>z2}. Since the set {y<0,y2>z2} is a cone and Rℓ(y,z) is for all ℓ∈[0,332] a homogeneous polynomial of degree 5, it only remains to check that
[TABLE]
We find that s=1 and s=−1 are roots of Rℓ(−1,s) for all ℓ∈[0,332], which allows us to consider
This motivates checking solutions of Nℓ(s)=0. We get
[TABLE]
We will show that M(s):=s(s2+3)3s2+1∈[0,1] for all s∈(−1,1), which implies that there exists no pair (ℓ,s)∈[0,332]×(−1,1), such that Nℓ(s)=0. Since N0(1)=32>0, this will then shows that Nℓ(s)>0 for all (ℓ,s)∈[0,332]×(−1,1) and in particular imply (4.37). We see that
[TABLE]
which implies that we can reduce our studies to s∈[0,1). The first derivative of M(s) is easily seen to fulfil
Hence, the equation M(s)=1 has no solutions in the half-open interval [0,1). We conclude that (4.37) holds true.
Summarising, we have proven that for all ℓ∈[0,332], H˘ℓ is a CCPSR manifold of dimension 2, which implies the same statement for Hℓ. This finishes the proof of Theorem 4.15.
∎
Now we have all necessary results at hand to complete the proof of our main result Theorem 1.1.
The existence of h∈Cn follows from Proposition 3.1 and Theorem 4.15. The statement h∈∂Cn={x3−x⟨y,y⟩+P3(y)∥z∥=1maxP3(z)=332} if and only if the initial H does not have regular boundary behaviour follows from Lemma 4.5 and Theorem 4.12.
The convexity of Cn follows from
[TABLE]
It remains to show that Cn⊂{x3−x⟨y,y⟩+P3(y)P3∈Sym3(Rn)∗}⊂Sym3(Rn+1)∗ is compact and that ∂Cn⊂Sym3(Rn+1)∗ is a continuous submanifold. For compactness we need to show that the condition ∥z∥=1maxP3(z)≤332 automatically implies that P3(⋅,⋅,⋅) viewed as a symmetric 3-tensor is bounded entry-wise, and we need to show that Cn is closed in the subspace topology101010With respect to the topology induced by the linear homeomorphy of Sym3(Rn+1)∗ and R6n3+6n2+11n+6. Note that said topology on Sym3(Rn+1)∗ does not depend on the choice of the linear homeomorphism.. This is equivalent to showing that all third derivatives of P3(z) are bounded on {∥z∥=1}. This follows from the fact that for all P3 fulfilling ∥z∥=1maxP3(z)=332, the corresponding h=x3−x⟨y,y⟩+P3(y)∈Cn defines a CCPSR manifold and, hence, we can use Corollary 4.4 and Proposition 4.9 to conclude that each entry in P3(⋅,⋅,⋅) is indeed bounded. Cn being closed follows from the continuity of ∥z∥=1maxP3(z) with respect to the prefactors of the monomials in P3(y), or equivalently the prefactors in the corresponding symmetric 3-tensor P3(⋅,⋅,⋅). We conclude that Cn⊂Sym3(Rn+1)∗ is compact in the subspace topology. The fact that ∂C is a continuous hypersurface in Sym3(Rn+1)∗ also follows from the continuity of the map P3↦∥z∥=1maxP3(z). However, note that this map is for n≥2 in general not smooth, or even differentiable.
To see this, consider the one-parameter family P3t(y)=y13+ty23, t∈[0,332], in Sym3(Rn+1)∗ and observe that
[TABLE]
does depend only continuously on t and is not continuously differentiable at t=1.
∎
Remark 4.16**.**
Note that for any compact set C⊂Sym3(Rn+1)∗
that contains an open neighbourhood of [math],
and any given CCPSR manifold H⊂{h=1}, we can always choose r>0, such that rh∈C. Then H is equivalent to r−31⋅H⊂{rh=1}. This shows that one can choose a generating set for the moduli set of n-dimensional CCPSR manifolds that is contained in a compact set C and, hence, bounded. It was however until now for n≥2 not known whether one can choose a compact generating set, like Cn in Theorem 1.1. For n=1 it was already shown in [CHM, Cor. 4] that the moduli set of CCPSR curves is generated by the set {x2y,x(x2−y2)}⊂Sym3(R2)∗, which is a compact set. One can show that x2y is equivalent to x3−xy2+332y3. By comparing with C1={x3−xy2+Ly3∣L∣≤332}, we see that x(x2−y2)=x3−xy2 is an inner point of C1 and x3−xy2+332y3 is one of the two points in ∂C1.
Theorem 1.1 in particular implies the following property of the moduli set of n-dimensional CCPSR manifolds (cf. Definition 2.12).
Corollary 4.17**.**
For n∈N fixed, let h,h∈Cn and let H⊂{h=x3−x⟨y,y⟩+P3(y)=1}, respectively H⊂{h=x3−x⟨y,y⟩+P3(y)=1}, denote the corresponding CCPSR manifolds containing the point (xy)=(10). Then the smooth curve
[TABLE]
defines an n-dimensional CCPSR manifold Ht⊂{γ(t)=(1−t)h+th=1} as the connected component containing (10) for all t∈[0,1], and H0=H, H1=H.
Example 4.18**.**
Corollary 4.17 does not say anything about whether the CCPSR manifolds Ht are pairwise equivalent or not.
i)
For an example of a curve γ as in Corollary 4.17 where the CCPSR manifolds Ht are pairwise inequivalent consider the family f) in Theorem 2.17 which corresponds to hyperbolic positive level sets of Weierstraß polynomials with positive discriminant [CDL, Lem. 1]. In Example 3.2, equation (3.16), we calculated the standard form for these polynomials and have seen that this family interpolates between the CCPSR surfaces Thm. 2.17 d) and e).
We see that after setting b=1−2t2 and swapping the coordinates y and z,
the family Thm. 2.17 f) is realized as a curve of CCPSR surfaces with
[TABLE]
connecting the CCPSR surface Thm. 2.17 d) corresponding to P3((yz))=0 respectively t=0, cf. equation (3.14), and the CCPSR surface Thm. 2.17 e) corresponding to P3((yz))=332y3 respectively t=1, cf. equation (3.17).
2. ii)
Next, consider P3((yz))=332y3+31yz2 and P3((yz))=332y3−32yz2. The corresponding CCPSR surfaces H and H are equivalent to Thm. 2.17 a) and Thm. 2.17 b), respectively, cf. Example 3.2 equations (3.11) and (3.12). One can further show that H is a homogeneous space with vanishing scalar curvature and H is a homogeneous space with scalar curvature given by −9/4. The curve γ as in Corollary 4.17 is given by
[TABLE]
The associated curve of CCPSR surfaces Ht contains precisely 3 inequivalent CCPSR surfaces. To show this, it suffices to show that for each t∈(0,1), Ht is equivalent to the CCPSR surface in Thm. 2.17 e) which coincides by Example 3.2 equation (3.15) with H1/2. Consider
[TABLE]
We see that (t,0)T∈dom(H1/3) for all r∈(−23,3). Next, we construct a linear transformation A(p) of the form (3.6) for p=p(r)=3h1/3((1,r,0)T)1(1r0). We leave it as an exercise for the reader to show that with appropriately chosen E(p) in (3.6),
[TABLE]
Since r∈(−3/2,3), this implies that H1/3 is indeed equivalent to Ht for all t∈(0,1), in particular to H1/2.
Note that Example 4.18 shows that the moduli set of CCPSR surfaces when equipped with the topology induced by the GL(3)-action is not Hausdorff. We expect this to be true in all dimensions, and also for CCGPSR manifolds of degree τ≥4.
5 Further applications
We will now demonstrate how to use Theorem 1.1 to obtain global properties of CCPSR manifolds. The first application is the existence of curvature bounds for the scalar curvature and the sectional curvature:
Corollary 5.1**.**
For any fixed dimension n≥2, there exist l(n),u(n),l(n),u(n)∈R, such that l(n)≤SH≤u(n) and l(n)≤KH≤u(n) for any n-dimensional CCPSR manifold H. Here SH denotes the scalar curvatures of H and KH denotes all possible sectional curvatures of H.
Proof.
Fix n≥2 and let H be a CCPSR manifold in standard form of dimension n. By Proposition 3.9, equation (3.30), we know that SH((10)) depends continuously on P3 and, hence, together with Theorem 1.1 we find that the following expressions are well defined real numbers:
[TABLE]
Proposition 3.1 now implies by the property that we can find a standard form of H with respect to any point p∈H that the so defined l(n),u(n)∈R fulfil l(n)≤SH(p)≤u(n) for all p∈H.
For the sectional curvatures, the proof proceeds analogously by using equation (3.39) in Lemma 3.11.
∎
The study of curvature bounds will be the main topic of an upcoming paper, in which we will expand our studies to manifolds in the image of the supergravity r- and q-map. Recall that, as mentioned in the introduction, sectional curvature bounds are the subject of active study in questions regarding the geometry of Kähler cones of Calabi-Yau threefolds [W, TW, KW].
Next we will give a proof that all closed PSR manifolds H equipped with their centro-affine fundamental form gH are geodesically complete. This was first shown in [CNS, Thm. 2.5]. In the corresponding proof it was used that the moduli set of closed PSR curves under the action of GL(2), which consists precisely of two elements, is compact [CHM, Cor. 4].
Note that geodesically complete PSR manifolds are necessarily closed, since otherwise we could always continuously extend gH to each boundary point and construct a geodesic in H which reaches said point in finite time, cf. [CNS, Prop. 2.4].
We will use the compactness property of Cn in Theorem 1.1 and the following lemma.
Lemma 5.2**.**
Let M be manifold of dimension n≥1 with a locally finite atlas, C⊂RN be a compact subset for some N∈N, and g(⋅):C→Γ(Sym2(T∗M)), c↦g(c), be a family of Riemannian metrics depending continuously on c∈C in the sense that the map
[TABLE]
is continuous.
Let p∈M be arbitrary and fixed. We denote by Brg(c)(p)⊂M the geodesic ball of radius r>0 around p∈M with respect to the Levi-Civita connection of g(c). Then the following is true:
[TABLE]
Proof.
Suppose (5.1) is false. Then there exists a sequence {ci,i∈N}⊂C, such that
[TABLE]
Since C⊂RN is compact, we can restrict to a subsequence if necessary and assume without loss of generality that {ci,i∈N} converges to a point c:=i→∞limci in C. Then, by assumption,
[TABLE]
But this is a contradiction to the fact that g(c) is a Riemannian metric and, hence, around every p∈M there exists a positive maximal radius r>0, such that Brg(c)(p)⊂M is compactly embedded (recall that independent of the considered Riemannian metric on M, the induced metric topology coincides with the given topology on M). Hence, (5.1) holds true.
∎
Proposition 5.3**.**
Closed PSR manifolds are geodesically complete.
Proof.
Let H⊂Rn+1 be an n-dimensional PSR manifold that is closed in its ambient space. Assume without loss of generality that H is connected, i.e. that H is a CCPSR manifold. Using Theorem 4.15, we can further without loss of generality assume that H=HP3⊂{hP3=x3−x⟨y,y⟩+P3(y)=1} is the connected component that contains the point (xy)=(10)∈{h=1} and that P3∈{∥z∥=1maxP3(z)≤332}⊂Sym3(Rn)∗, where we view the set {∥z∥=1maxP3(z)≤332} as a compact subset of RN with N=dimSym3(Rn)∗=6n3+3n2+2n. Consider the set
[TABLE]
Lemma 4.3 implies that M={∥z∥<23}⊂Rn, in particular M is a smooth submanifold of Rn. Recall that with βP3(z):=hP3((1z)), (HP3,gHP3) is isometric to
[TABLE]
for all P3∈{∥z∥=1maxP3(z)≤332}, cf. (3.29). Since M⊂dom(HP3) independent of P3,
we can consider the family of Riemannian metrics on M
[TABLE]
Since g(⋅) depends continuously on the compact subset {∥z∥=1maxP3(z)≤332}⊂Sym3(Rn)∗
in the sense of Lemma 5.2 (where we identify Sym3(Rn)∗ with RN as above and note that M as an open submanifold of Rn is in particular equipped with a finite atlas consisting of a single chart), we can use Lemma 5.2 and obtain that there exists r>0, such that Brg(P3)(0)⊂M
is compactly embedded for all P3∈{∥z∥=1maxP3(z)≤332}. Together with Proposition 3.1 this implies that for all P3∈{∥z∥=1maxP3(z)≤332} and all p∈HP3, BrgHP3(p)⊂HP3 is compactly embedded. Now we use the fact that a Riemannian manifold (M,gM) is complete if and only if we can find r>0, such that BrgM(p) is compact in M for all p∈M, for a proof of that statement see [Li, Lem. 2.21].
We conclude that (HP3,gHP3) is complete for all P3∈{∥z∥=1maxP3(z)≤332}.
∎
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