Geometry and Topology of the space of plurisubharmonic functions
Soufian Abja

TL;DR
This paper introduces a new geometric space of strongly plurisubharmonic functions on pseudoconvex domains, analyzes its metric properties via Mabuchi geodesics, and explores applications to local Kähler-Einstein metrics.
Contribution
It defines the Mabuchi space for strongly plurisubharmonic functions and investigates its metric and regularity properties, especially in the ball, with applications to Kähler-Einstein metrics.
Findings
Established regularity of Mabuchi geodesics in the ball
Analyzed metric properties of the Mabuchi space
Studied existence of local Kähler-Einstein metrics
Abstract
Let be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in . We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local K\"ahler-Einstein metrics.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Advanced Differential Geometry Research
Geometry and Topology of the space of plurisubharmonic functions
Soufian ABJA
Ibn Tofail University
Kenitra Morocco
Abstract.
Let be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in . We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Einstein metrics.
Contents
Introduction
Let be a compact Kähler manifold and a Kähler class. The space of Kähler metrics in can be seen as an infinite dimensional riemannian manifold whose tangent spaces can all be identified with . Mabuchi has introduced in [Mab87] an -metric on , by setting
[TABLE]
where and denotes the volume of . Mabuchi studied the corresponding geometry of , showing in particular that it can formally be seen as a locally symmetric space of non positive curvature. The (geometry) metric study of the space has motiveted a lot of interesting works in the last decades, see notably [Don99, Chen00, CC02, CT08, Chen09, LV13, DL12, Dar13, Dar14, Dar15].
The purpose of this article is to extend some of these studies to the case when is a smooth strongly pseudoconvex bounded domain of . We note here that this problem of extension to the local case recently been considered by Rashkovskii [Rash16] and Hosono [Hos16], Rashkovskii studied geodesics for plurisubharmonic functions in the Cegrell class on a bounded hyperconvex domain, he also showed that functions with strong singularities generally cannot be connected by (sub)geodesic arcs. Hosono described the behavior of the weak geodesics between toric psh functions with poles at the origin.
Our first interest is the geometry of the space of plurisubharmonic functions, We equipped the space of plurisubharmonic functions with a Levi-Civita connection and we describe the tensor curvature and sectional curvature as in a paper of Mabuchi [Mab87]. Our first main result is to the establish that the space of plurisubharmonic functions is a locally symmetric space:
Theorem A. The Mabuchi space equipped with the Levi-Civita connection is a locally symmetric space.
Following the work of Donaldson [Don99] and Semmes [Sem92] in the compact setting, we reinterpret the geodesics as a solution to a homogeneous complex Monge-Ampère equation. Weak geodesics are introduced as an envelope of functions:
[TABLE]
Our second main result is to establish regularity properties of geodesics in the ball by adapting the celebrated result of Bedford-Taylor[BT76]:
Theorem B. * Let be the unit ball in . Let and be the end geodesic points which are . Then the Perron-Bremermann envelope*
[TABLE]
admits second-order partial derivates almost everywhere with respect to variable which locally bounded uniformly with respect to , i.e for any compact subset there exists which depend on and such that
[TABLE]
The existence of local Kähler-Einstein metrics was studied by Guedj, Kolev and Yeganefar [GKY13] in bounded smooth strongly pseudoconvex domains which are circled. This is equivalent to the resolution of the following Dirichlet problem
[TABLE]
They treated also the following family of Dirichlet problems
[TABLE]
showing that there is a solution for . We apply our study of the geodesics problem and an idea of [DR15, DG16] to prove that the existence of a solution to is equivalent to the coercivity of the Ding functional:
Theorem C. Let be a smooth strongly pseudo-convex circled domain. If there exists such that,
[TABLE]
*then admits a -invariant smooth strictly plurisubharmonic function solution.
Conversely if admits such a solution and is strictly -convex, then there exists such that,*
[TABLE]
The organization of the paper is as follows.
- •
Section 1 is devoted to preliminary results and the definition of the space and its geometry.
- •
In Section 2 we show that the geodesics are continuous (sometimes even Lipschitz) up to the boundary of .
- •
In section 3 we prove theorem B.
- •
Finally, we prove Theorem C in Section 4.
Acknowledgements**.**
It is a pleasure to thank my supervisors Vincent Guedj and Said Asserda, for their support, suggestions and encouragement. I thank Ahmed Zeriahi for very useful discussions and suggestions. Also, i would like to thank Tat Dat Tô and Zakarias Sjöström Dyrefelt for a very careful reading of the preliminary version of this paper and very useful discussions.
1. Mabuchi geometry in pseudoconvex domains
In this section we will study the geometry of the space of plurisubharmonic functions in strongly pseudoconvex domain, based upon works of Mabuchi [Mab87], Semmes[Sem92] and Donaldson [Don99], as it was clarified through lecture notes of Guedj [G14] and Kolev [Kol12].
1.1. Preliminaries
In this section we recall some analytic tools which will be used in the sequel. Let be a smooth pseudoconvex bounded domain. Recall that a bounded domain is strictly pseudoconvex if there exists a smooth function defined in neighbourhood of such that with , where
[TABLE]
Definition 1.1**.**
We let denote the set of plurisubharmonic functions in . In particular a function is , upper semi-continuous and such that
[TABLE]
in the weak sense of positive currents.
The following cone of ”test functions” has been introduced by Cegrell [Ceg98]:
Definition 1.2**.**
[Ceg98]** We let denote the convex cone of all bounded plurisubharmonic functions defined in such that , for every , and .
Definition 1.3**.**
[Ceg98]** The class is a set of functions for which there exists a sequence of functions decreasing towards in all of , and so that .
We will need the following maximum principle:
Proposition 1.4**.**
[BT76]** Let , be locally bounded plurisubharmonic functions in such that . Then
[TABLE]
1.2. The space of plurisubharmonic potentials
We begin this section by defining the Mabuchi space of plurisuharmonic functions in .
Definition 1.5**.**
The Mabuchi space of plurisubharmonic functions in is:
[TABLE]
We now consider the tangent space of in every .
Definition 1.6**.**
The tangent space of at point , we denote by is the linearisation of defined by:
[TABLE]
The tangent space of at can be identified with
[TABLE]
Indeed. Let , we put for close enough to [math] we have , and
[TABLE]
this implies that hence
[TABLE]
Conversely, let which gives for every . In particular , therefore
[TABLE]
Definition 1.7**.**
[Mab87]** The Mabuchi metric is the Riemanniann metric. It is defined by
[TABLE]
where .
1.3. Mabuchi geodesics
Geodesics between two points , in are defined as the extremals of the Energy functional
[TABLE]
where is a path in joining to . The geodesic equation is obtained by computing the Euler-Lagrange equation of the functional .
Theorem 1.8**.**
The geodesic equation is
[TABLE]
where is the gradient relative to the metric .
Proof.
We need to compute the Euler-Lagrange equation of the Energy functional. Let be a variation of with fixed end points,
[TABLE]
Set and observe that and on . Thus
[TABLE]
and
[TABLE]
A direct computation yields
[TABLE]
Integration by part, and the fact yields
[TABLE]
And we have also by Stokes and the fact on
[TABLE]
hence
[TABLE]
which implies
[TABLE]
Therefore is critical point of if and only if
[TABLE]
∎
1.4. Levi-Civita connection
As for Riemanniann manifolds of finite dimension. One can find the local expression of the Levi-Civita connection by polarizing the geodesic equation.
Definition 1.9**.**
We define the covariant derivative of the vector field along the path in by the formula
[TABLE]
Theorem 1.10**.**
* is the Levi-Civita connection.*
Proof.
To show that is a Levi-Civita connection, we must show that the connection is metric-compatible and a torsion-free.
i) Metric-compatibility: Let , be two vector fields
[TABLE]
(The passage from the second line to the third line is a result of the equation
[TABLE]
and Stokes theorem).
ii) is a torsion-free, because
[TABLE]
Thus is a Levi-Civita connection. ∎
1.5. Curvature tensor
We will define the curvature tensor and the sectional curvature and we will give those expressions. We will finish by proving that the space of plurisubharmonic functions is locally symmetric. We start by giving some definitions and conventions.
Definition 1.11**.**
Let and be two functions in the tangent space of at . The Poisson bracket of and compared to the form is
[TABLE]
where is the inverse matrix of .
Lemma 1.12**.**
Let , and three functions belonging to the tangent space of at . The Poisson bracket satisfies the following properties :
i)
ii)
iii)
iv)
v)
*vi) .
where and is the Lie bracket.
Let be a function in tangent space, the Hessian of is defined by , where is the Levi-Civita connection respectively to the form . We Recall in the next lemma some proprieties of the Hessian well know in the literature.
Lemma 1.13**.**
*Let and be two vector fields. Then the Hessian satisfies the following proprieties:
i) .
ii) .
iii)
Where and are the Levi-Civita connection and the metric respectively associated to the form .*
In the sequel of this section, we consider a 2-parameters family and a vector field defined along . We denote by
[TABLE]
Definition 1.14**.**
The curvature tensor of the Mabuchi metric in is defined by
[TABLE]
*where is 2-parameters family and vector field .
The sectional curvature is given by*
[TABLE]
Theorem 1.15**.**
The curvature tensor of the Mabuchi metric in can be expressed as
[TABLE]
The sectional curvature is the following
[TABLE]
where is the Poisson bracket associate to the form .
Proof.
To compute the curvature tensor of , we compute the first term in the definition of the curvature tensor . Indeed, let be the vector field, its derivative along the path
[TABLE]
where
[TABLE]
we derive the along the path as follows
[TABLE]
We express the second term in RHS of the last equation:
[TABLE]
By applying of the three properties of lemma 1.13 by taken and , then we express the last term in the last equation as follows:
[TABLE]
Which gives
[TABLE]
We develop the fourth term in the RHS in the last equation by applying the second proprieties of the lemma 1.13, taken and :
[TABLE]
We have also by applying the first proprieties of lemma 1.13:
[TABLE]
Where . Then we have:
[TABLE]
After all previous equations, we get the expression of as follows:
[TABLE]
We get the expression of by reversing the roles of and as follows:
[TABLE]
Therefore we get
[TABLE]
In the line three we use the fact that the Levi-Civita connection is torsion free. In the line four we use the fourth property in lemma 1.12, in the last line we use the second property in lemma 1.12. We calculate the sectional curvature as follow:
[TABLE]
We use in line three the expression of the curvature tensor, in the line four we use fifth property in lemma 1.12. ∎
Definition 1.16**.**
We say a connection in is locally symmetric if its curvature tensor is parallel i.e .
Theorem 1.17**.**
The Mabuchi space provided by the Levi-Civita connection is a locally symmetric space.
Proof.
Let be 3-parameters family in .
[TABLE]
We use the expression of the curvature tensor and the sixth property in the lemma 1.12 of the Poisson bracket. Therefore
[TABLE]
hence is locally symmetric. ∎
2. The Dirichlet problem
We now study the regularity of geodesics using pluripotential theory, the tools using are developed by Bedford and Taylor [BT76, BT82].
2.1. Semmes trick
We are interested in the boundary value problem for the geodesic equation: given , two distinct points in , can one find a path in which is a solution of(1) with end points and ? For each path in , we set
[TABLE]
We will show in this section that the geodesic equation in is equivalent to Monge-Ampère equation on as in Semmes [Sem92].
Lemma 2.1**.**
The Monge-Ampère measure of the function in is :
[TABLE]
with
[TABLE]
Proof.
We write and , and we give also the expression of in . Indeed
[TABLE]
with such that . Then we can find the expression of in . Indeed
[TABLE]
On the second line we use Leibniz formula and the fact that on the third line. ∎
Theorem 2.2**.**
* is a geodesic if and only if .*
Proof.
From the previous lemma, we have
[TABLE]
The first term in RHS of the last equation equal to [math] a cause of bi-degree. We have
[TABLE]
and
[TABLE]
and we have also , which gives
[TABLE]
and
[TABLE]
Now we can explain the second term also. Indeed
[TABLE]
And also for third term, we have
[TABLE]
After the previous equations we have,
[TABLE]
From the fact that , we infer that is geodesic if and only if
[TABLE]
∎
After the previous theorem we deduce that the geodesic problem in Mabuchi space is equivalent to the following Dirichlet problem:
[TABLE]
2.2. Continuous envelopes
We have that and are smooth, in the sequel we can assume that and are only .
Definition 2.3**.**
The Perron-Bremermann envelope is defined by
[TABLE]
with
[TABLE]
*Where and \Psi_{\partial A\times\Omega}=\left\{\begin{array}[]{ll}\varphi_{0}(z)&\hbox{{|\zeta|=1};}\\ \varphi_{1}(z),&\hbox{{|\zeta|=e}.}\end{array}\right.
.*
Theorem 2.4**.**
*If . Then the Perron-Bremermann envelope satisfies the following conditions:
i) .
ii) .
iii) in .*
Proof.
Let be a strictly plurisubharmonic defining of . Observe that the family is not empty .
i)We start by proving the plursubharmonicity of in . We can write the Dirichlet problem on following way:
[TABLE]
with . Let be a harmonic function in , continuous up to the boundary of , the solution of the following Dirichlet problem
[TABLE]
Exists, since is a regular domain.
For all , we have on ,which implies
[TABLE]
Furthermore we have
[TABLE]
Then by maximum principle
[TABLE]
the last inequality holds for every function in , hence it holds for upper envelope of subsolution
[TABLE]
It also holds for its upper semi-continuous regularization on the boundary , we get
[TABLE]
and consequently
[TABLE]
Since the function is plurisubharmonic in and
[TABLE]
we infer that
[TABLE]
hence is plurisubharmonic function in .
Since is plurisubharmonic in , implies is upper semi continuous. We now prove it is lower upper semi-continuous. Indeed Fix and since is compact and the function is continuous on , we can choose so small that
[TABLE]
Fix with . So, We have the following inequality
[TABLE]
and
[TABLE]
It follows that the function
[TABLE]
is plurisubharmonic in because
-
If it coincides with which is plurisubharmonic.
-
If , it is the maximum of two plurisubharmonic functions .
-
After the two previous inequalities, we infer that the function coincides on the boundary, furthermore
[TABLE]
Which implies , finally we get
[TABLE]
Thus is lower semi-continuous, therefore it is continuous .
ii)We are going to prove that
[TABLE]
Firstly, since we have
[TABLE]
To prove the reverse of inequality, we construct a plurisubharmonic barrier function at each point . Since is strictly plursubharmonic function, we can choose large enough so that the function
[TABLE]
is plurisubharmonic in and continuous up to the boundary such that with for all .
Fix and take such that and . We choose a big constant so that
[TABLE]
This implies that the function is
[TABLE]
Thus we have which implies in . We get
[TABLE]
therefore
[TABLE]
iii) The Perron Bremermann envelope
[TABLE]
is plurisubharmonic continuous up the boundary of and .
By a lemma due to Choquet, this envelope can be realised by a countable family
[TABLE]
We put
[TABLE]
the function is increasing and
[TABLE]
Let be any ball, we consider the following Dirichlet problem
[TABLE]
since
[TABLE]
and
[TABLE]
we have
[TABLE]
We consider the following function
[TABLE]
The function belongs to . This implies
[TABLE]
furthermore
[TABLE]
then
[TABLE]
therefore
[TABLE]
since is arbitrary we give
[TABLE]
By the continuity property of Monge-Amèpre operators of Bedford and Taylor along monotone sequences, we have
[TABLE]
i.e
[TABLE]
∎
2.3. Lipschitz regularity
In this subsection we will give the geodesic regularity Lipschitz in time and in space. We begin by regularity Lipschitz in time. We use a barrier argument as noted by Berndtsson [Bern15]
Proposition 2.5**.**
The Perron Bremermann envelope is Lipschitz function with respect to .
Proof.
The proof follows from a classical balayage technique. Indeed, we consider the following function
[TABLE]
where is a big constant. Furthermore
[TABLE]
The last line follows by on and . Then it belongs to and
[TABLE]
since and , which implies
[TABLE]
[TABLE]
which gives , similarly for other case . Since the function is convex along (by subharmonicity in ),we infer that for almost everywhere ,,
[TABLE]
then is uniformly Lipschitz at . ∎
We will prove the regularity Lipschitz in space by adapting the method of Bedford and Taylor [BT76](see also [GZ17]).
Theorem 2.6**.**
The Perron Bremermann envelope is Lipschitz function up to the boundary with respectively to space variable.
Proof.
Let be a smooth defining of which is strictly psh in a neighbourhood of , and also be a smooth defining of which is strictly psh in a neighbourhood of . We will construct an extension of function defined on the boundary of by
[TABLE]
Let be a smooth function with compact support defined in by near of [math] and by near of . We put
[TABLE]
is a smooth function in . We have near of and near of .
We consider the following function:
[TABLE]
where , , The function satisfies
[TABLE]
The function is extension plurisubharmonic of the function defined in to . We can also extend the function defined in by putting
[TABLE]
where is a big constant.
On two cases the function satisfies the following properties
[TABLE]
By maximum Principle we get
[TABLE]
Applying the same process to the boundary data we choose function defined in such that on , the maximum Principle implies
[TABLE]
After the two previous inequalities we have
[TABLE]
Since in , the envelope can be extended respectively to variable as a plurisubharmonic function in by setting in with fixed in . Fix so small that whenever and , this set noted in sequel by . Fix such that . Recall that and are Lipschitz in each variable, thus
[TABLE]
for any and .
Observe that the function is well defined psh in the open set . If and , then
[TABLE]
If and , then
[TABLE]
This shows that the function defined by
[TABLE]
is plurisubharmonic in . Since on we get in , hence in . We have shown that
[TABLE]
whenever , and . Replacing by shows that
[TABLE]
Which proves that is Lipschitz in every . ∎
3. Case of the unit ball
In this section we shall show how to use the proof of Bedford and Taylor [BT76], which is simplified by Demailly [Dem93] in the unit ball for giving the regularity in space variable for our geodesics problem. We need some preparation for prove this regularity. The open subset of is called the unit ball. First we shall define the Mobius transformation of the unit ball. Let . Denote the orthogonal projection onto the subspace of in generated by the vector by,
[TABLE]
The Mobius transformation associated with a is the mapping
[TABLE]
With denote the hermitian scalar product of and . For every , the Mobius transformation has the following properties
i) and .
ii)an elementary computation yields
[TABLE]
with and is uniformly with respect of .
We need in the sequel the following useful lemma for giving the regularity in unit ball.
Lemma 3.1**.**
Let be a plurisubharmonic function in domain , assume that there exists such that
[TABLE]
and for all and . Then is -smooth , ant its second derivative, which exists almost everywhere, satisfies
[TABLE]
Proof.
Let denote the standard regularization of defined in for . Fix small enough and . Then for we have
[TABLE]
It follows from Taylors formula that if
[TABLE]
therefore by having for all and . Now for
[TABLE]
Recall that is plurisubharmonic in hence
[TABLE]
The above upper-bound also yields a lower-bound of
[TABLE]
For any and . This implies that
[TABLE]
Thus, we have shown that is uniformly Lipschitz in . We infer that is Lipschitz in and uniformly in compact subsets of . Since the dual of is , it follows from the Alaoglu-Banach theorem that, up to extracting a subsequence,there exists a bounded function such that weakly in . Now in the sense of distributions hence . Therefore is in and its second-order derivative exists almost everywhere with . ∎
Theorem 3.2**.**
Let be the unit ball in . Let and be the end geodesic points which are . Then the Perron-Bremermann envelope
[TABLE]
admits second-order partial derivates almost everywhere with respect to variable which locally bounded uniformly with respect to , i.e for any compact subset there exists which depend on and such that
[TABLE]
Proof.
For proving the theorem, we weed to prove the following inequality
[TABLE]
for any , and .
The idea is to study the boundary behavior of the plurisubharmonic function in order to compare it with the function in . This does not make sense since the translations do not preserve the boundary. We are instead going to move point by automorphisms of the unit ball: the group of holomorphic automorphisms of the latter acts transitively on it and this is the main reason why we prove this result for the unit ball rather than for a general strictly pseudoconvex domain (which has generically few such automorphisms).
By the fact is Lipschitz with respectively to variable (theorem 2.6) and expansion (2) we have
[TABLE]
and
[TABLE]
which implies
[TABLE]
We consider the following functions:
[TABLE]
and , we Observe that the functions and are well defined in and plurisubharmonic in . We need to show that
[TABLE]
For showing the last inequality we will apply the maximum Principle, then we need to prove
[TABLE]
and
[TABLE]
The last inequality is easy follows from the fact that is a plurisubharmonic and in by (theorem 2.4).
We need to compare and in the boundary of . Indeed, since , then we will compare in two parts, we begin by the part , in this part we get
[TABLE]
For shows that , we take just
[TABLE]
For the second part , We compare in only ,because belongs to the previous part, since , we begin this part by comparing in case , we have
[TABLE]
and
[TABLE]
We apply Taylor expansion and we get
[TABLE]
and
[TABLE]
Which is implies
[TABLE]
where depend only on the then
[TABLE]
If we take , we get on . By same methods we get on for , where depend only on the which concludes the second part.
By part one and two we infer
[TABLE]
From the maximum Principle we get
[TABLE]
Which is implies
[TABLE]
Observe that the mapping is a local diffeomorphism in neighborhood of the origin as long as , which depend on smoothly and its inverse is linear with a norm less than or equal to since
[TABLE]
which gives
[TABLE]
Fix a compact set compact, there exists such that and we have
[TABLE]
after the previous lemma we get
[TABLE]
with . ∎
4. Moser-Trudinger inequalities
In this section we assume is pseudoconvex circled domain. We try to solve the complex Monge-Ampère equation
[TABLE]
with smooth and plurisubharmonic, and just the Lebesgue normalised so that . We know that
- •
We can solve this equation if is not too large ( is treated in [GKY13] and even ).
- •
One can not solve the equation if is too large, cf [GKY13, section 6.2] and [BB11].
We denote by
[TABLE]
the Monge-Ampère energy functional of a plurisubharmonic function , which is defined as the primitive of Monge-Ampère operator. The expression
[TABLE]
defines the Ding functional.
Definition 4.1**.**
We say the functional is coercive , if there exist and such that :
[TABLE]
Definition 4.2**.**
Set . The continuous family is called the geodesic joining and .
We show that is linear along of geodesics, this result is in [GKY13, lemma 22], and was proven by Rashkovskii [Rash16] in the Cegrell class, we reprove it for continuous geodesics for convenience of the reader.
Lemma 4.3**.**
Let be a continuous geodesic. Then is affine.
Proof.
After the proof of theorem 2.2 we have
[TABLE]
We have by definition of
[TABLE]
Which implies
[TABLE]
[TABLE]
where the second equality follows from Stokes theorem because on , and the last one be above calculation.
Thus, it follows from theorem 2.4 that is harmonic in . Since is invariant by rotation with respect to , hence it is affine in . ∎
We recall here [GKY13, proposition 23].
Proposition 4.4**.**
Assume that is circled, let be an -invariant solution of . Then
[TABLE]
where denotes all -invariant plurisubharmonic functions in which are continuous up to the boundary, with zero boundary value.
Proof.
Let be a geodesic joining to . It follows from work of Berndtsson [Bern06] that
[TABLE]
is convex, since is affine from lemma 4.3. Then is concave.
therefore it is sufficient to show that the derivative of at is non-negative to conclude for all s, in particular at where it yields . When is smooth, a direct computation yields, for ,
[TABLE]
For the general case, the same method as in the proof of [BBGZ13, theorem 6.6] applies. ∎
Lemma 4.5**.**
The Functional is upper semi-continuous in .
Proof.
Recall . The first term is upper semi-continuous in . For the second term we apply Skoda uniform integrability theorem[Zer01].
Assume without loss of generality that . We need to check that is upper semi-continuous.
Let be a sequence in converging to these functions have zero Lelong number. The following extension:
to as in , in . We apply Skoda’s uniform integrability estimates:
[TABLE]
[TABLE]
as follows from the Cauchy-Schwarz inequality and the elementary inequality
[TABLE]
The conclusion follows since converges to in . ∎
We recall that the Dirichlet problem has a solution for by [GKY13], we moreover have uniqueness if is stricltly -convex( is strictly convex dor the metric ). We recall here the main result of [GKY13].
Theorem 4.6**.**
Let be a bounded smooth strongly pseudoconvex domain which is circled. Let be a smooth -invariant strictly plurisuharmonic solution of the complex MongeAmpère problem . If is strictly -convex, then is the unique -invariant solution of .
Inspired by Dinezza-Guedj [DG16, theorem 5.5], we now prove the following theorem
Theorem 4.7**.**
Let be a smooth strongly pseudo-convex circled domain. If there exists such that,
[TABLE]
*then admits a -invariant smooth strictly plurisubharmonic function solution.
Conversely if admits such a solution and is strictly -convex, then there exists such that,*
[TABLE]
Proof.
If we assume the following inequality holds,
[TABLE]
then the same method of [GKY13]applies, if only we change by .
Conversely, as is a solution of then from the (proposition 4.4) we have
[TABLE]
assume for contradiction that there is no such that
[TABLE]
for all . Put and . Then we can find a sequence such that
[TABLE]
We discuss here two cases, the first case if does not blow up to , we reach a contradiction, by letting go to . Indeed we can assume that bounded and converges to some which is -invariant. Since is upper semi-continuous by lemma 4.5, we infer contradiction because is the solution of .
The second case if . It follows that .
We let denote the weak geodesic joining to and set . We know that is is affine along of the Mabuchi geodesic. Thus , where and are real numbers. For we have
[TABLE]
and for we have
[TABLE]
therefore . Then
[TABLE]
Since is affine along of the Mabuchi geodesic and by Berndtsson [Bern06] convexity result, we infer that the map is concave, which implies with (5) that
[TABLE]
thus . This shows that is a maximizing sequence for . If we take on equation (6) we get
[TABLE]
Passing to subsequence, we can assume that converge to which is -invariant. Since is upper semi-continuous and is a maximizing sequence for then we have and so thanks to the uniqueness. Letting to infinity in (7) we get
[TABLE]
This yields a contradiction. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[BT 82] E. Bedford, B. A. Taylor: A new capacity for plurisubharmonic functions. Acta Math. 149 (1982), no. 1-2, 1–40.
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