On Action of $PSL(n+1, {\bf C})$ on Space of $L_k^p$-maps on ${\bf P}^n$

TL;DR

This paper proves the properness of the action of the projective special linear group on the space of certain stable maps on complex projective space, advancing understanding of symmetries in geometric analysis.

Contribution

It establishes the properness of the group action on the space of $L_k^p$-maps, providing new insights into the structure of these moduli spaces.

Findings

Properness of the $PSL(n+1, f C)$ action on $L_k^p$-maps

Results on $v$-stability of maps

Implications for geometric analysis and moduli space structure

Abstract

In this paper, we prove properness of the action of the reparametrization group $PSL(n+1, {\bf C})$ on the space of $v$-stable $L_k^p$-maps on ${\bf P}^n$ as well as related results.

Equations

$$v(f\circ u{|}_{R_1})={\int }_{R_1}{\tilde{v}}_m(d(f\circ u))={\int }_{R_1}{u}^{\ast }({\tilde{v}}_m(df))$$

$$={\int }_{u(R_1)}(u^{-1}{)}^{\ast }[u^{\ast }({\tilde{v}}_m(df))]={\int }_{u(R_1)}{\tilde{v}}_m(df)=v(f{|}_{u(R_1)}).$$

$$v_{m-1,\beta }(f{|}_{{\cup }_{i=1}^{l}U(S(x_i))})={\Sigma }_{i=1}^l|{\int }_{S^{\prime }(x_i)}{f}^{\ast }({\alpha }_{i}d{\tilde{\nu }}_{{\tilde{S}}^{\prime }(x_i)})|$$

$$={\Sigma }_{i=1}^l|{\int }_{S^{\prime }(x_i)}{f}^{\ast }({\alpha }_i)\cdot {\tilde{v}}_{m-1}(df{|}_{S^{\prime }(x_i)})|.$$

$$v_m((h_i\circ g_i){|}_{B(x_1;{\rho }_1)})=v_m(h_i{|}_{g_i(B(x_1;{\rho }_1))})$$

$$\le ||h_i|{|}_{C^1}^{2n}vol(g_i(B(x_1;{\rho }_1)))\le C_4\cdot ||h_i|{|}_{C^1}^{2n}{ϵ}^2$$

$$\le C_4||h_i|{|}_{k,p}^{2n}{ϵ}^2\le C_4(\Vert f_1{\Vert }_{k,p}+||h_i-f_1|{|}_{k,p}{)}^{2n}{ϵ}^2$$

$$\le C_4(\Vert f_1{\Vert }_{k,p}+{ϵ}_1{)}^{2n}{ϵ}^2.$$

$$vo{l}_{m-1}({\Delta }^{\ge K}(a_i)(v_i(B_{m-1}(x_1;{\rho }_1))))\le M\cdot vo{l}_{K-1}(E_i).$$

$$vo{l}_{m-1}(\Delta (a_i)(v_i(B_{m-1}(x_1;{\rho }_1))))$$

$$=vo{l}_{m-1}({\Delta }^{<K}(a_i)\circ {\Delta }^{\ge K}(a_i)(v_i(B_{m-1}(x_1;{\rho }_1))))$$

$$\le M\cdot vo{l}_{K-1}({\Delta }^{<K}(a_i)(E_i)).$$

$$v_{m-1}((h_i\circ g_i){|}_{B_{m-1}(x_1;{\rho }_1)})=v_{m-1}(h_i{|}_{g_i(B_{m-1}(x_1;{\rho }_1))})$$

$$\le ||h_i|{|}_{C^1}^{2n-1}vo{l}_{m-1}(g_i(B_{m-1}(x_1;{\rho }_1)))$$

$$\le C(\Vert f_1{\Vert }_{k,p}+||h_i-f_1|{|}_{k,p}{)}^{2n-1}vo{l}_{m-1}(g_i(B_{m-1}(x_1;{\rho }_1)))$$

$$\le C(\Vert f_1{\Vert }_{k,p}+{ϵ}_1{)}^{2n-1}vo{l}_{m-1}(g_i(B_{m-1}(x_1;{\rho }_1))).$$

Full text

1 Introduction

In [Liu 2013, Liu 2015] we have introduced the notion of weak-stable nodal $L_{k}^{p}$-maps as a natural generalization of stable $J$-holomorphic maps in Gromov-Witten theory introduced by Kontsevich in [Kontsevich 1995]. The purpose of this paper is to generalize part of the results on proper action of the reparametrization groups on the space of stable or weakly stable $L_{k}^{p}$-maps [Liu 2013, Liu 2015] to the case with domain ${\bf P}^{n}$ acting on by $PSL(n+1,{\bf C})$. Further generalizations to other higher dimensional but smooth domains will be treated in a companion of this paper.

The main difficulty for such a generalization lies on the well-known fact: unlike $1$-dimensional case, higher dimensional biholomorphic maps are not conformal in general. Thus even though a version of higher dimensional energy function of the $L_{k}^{p}$-maps, the key quantity used for the $1$-dimensional case [Liu 2015], can be defined, it is only invariant with respect to the conformal automorphisms of the domains but not respect to the natural action of the reparametrization group of biholomorphic automorphisms.

In this paper, instead of using energy function, the volume function $v(f)$ is used to define the notion of $v$-stability for $L_{k}^{p}$-maps and to prove part of the corresponding results in [Liu 2015] described as follows.

Let $M$ and $N$ be compact Riemannian manifolds of class $C^{\infty}$ with $\dim(M)\leq\dim(N)$ and ${{\cal M}}_{k,p}(M,N)$ be the mapping space of $L_{k}^{p}$-maps. Throughout the paper, we always assume that $m_{0}=[k-\frac{m}{p}]\geq 1$ where $m=dim(M)$ so that each $L_{k}^{p}$-map is at least of class $C^{1}$ by Sobolev embedding theorem.

When the domain $M$ is ${\bf P}^{n}$, the group $G=PSL(n+1,{\bf C})$ operates on the space ${{\cal M}}_{k,p}({\bf P}^{n},N)$ as the group of reparametrizations.

Recall in the case that $X$ is a locally compact topological space such as a finite dimensional manifold and $G$ is a Lie group or a locally compact topological group, a (continuous) group action $\Phi:G\times X\rightarrow X$ is said to be proper if the map $\Phi\times{\it id}_{X}:G\times X\rightarrow X\times X$ is a proper map: the inverse image $(\Phi\times{\it id}_{X})^{-1}(K)$ of any compact subset $K\subset X\times X$ is compact.

In our infinite dimensional case, ${{\cal M}}_{k,p}({\bf P}^{n},N)$ is not locally compact, the above definition is too weak to be useful.

In [Liu 2013, Liu 2015], we have introduced the following stronger definition.

Definition 1.1

A group action $\Phi:G\times X\rightarrow X$ is said to be proper if for any compact subset $K\subset X\times X$, there is a neighborhood $U$ of $K$ in $X\times X$ such that the image $\pi_{G}((\Phi\times{\it id}_{X})^{-1}(U))$ of the projection to $G$ of the inverse image $(\Phi\times{\it id}_{X})^{-1}(U)$ is pre-compact in $G$.

It was proved in [Liu 2013, Liu 2015] that in the case $X$ is a locally compact topological space acting by a locally compact group $G$, the definition here is equivalent to the usual one.

Now we define the notion of $v$-stability [Liu 2013].

Definition 1.2

A $L_{k}^{p}$-map $f:M\rightarrow N$ with $dim(M)=m$ is said be to $v$-stable if its volume $v(f)>0$. Here $v(f)=v_{m}(f)$ if $v_{m}(f)>0$ and $v(f)=v_{m-1}(f)$ if $v_{m}(f)=0$, defined by using the two volume functions $v_{m-1}$ and $v_{m}$ of dimension $m-1$ and $m$ respectively.

The notion of $v_{m-1}$/ $v_{m}$-stability is defined similarly.

Remark 1.1

In the case that $dim(M)=2$, $f:M\rightarrow N$ is $v$-stable if and only if $f$ is not a constant map, and hence weakly stable by the definition in [Liu 2015]. In this sense $v$-stability is the natural generalization of the notion of weak stability in [Liu 2015] to the higher dimensional case with smooth domains. The theorems below show that $v$-stable maps have the similar properties as weakly stable maps.

Let ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ be the space of $v$-stable $L_{k}^{p}$-maps on ${\bf P}^{n}$.

Theorem 1.1

The action of $G=PSL(n+1,{\bf C})$ on the space ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ is proper.

Corollary 1.1

For any $v$-stable $L_{k}^{p}$-map $f$, its stabilizer $\Gamma_{f}$ is always a compact subgroup of $G$.

Proposition 1.1

The action of $G=PSL(n+1,{\bf C})$ on the space ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ is $G$-Hausdorff. Therefore, the quotient space ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)/G$ of unparametrized $v$-stable $L_{k}^{p}$-maps is Hausdroff.

Corollary 1.2

Given any $f$ in ${{\cal M}}^{*}_{k,p}$, the G-orbit $G\cdot f$ is closed in ${\cal M}^{*}_{k,p}$.

Definition 1.3

A $L_{k}^{p}$-map $f$ in ${{\cal M}}^{*}_{k,p}$ is said to be stable if its stabilizer $\Gamma_{f}$ is a finite group.

Theorem 1.2

Any $v_{2n}$-stable $L_{k}^{p}$-map on ${\bf P}^{n}$ is stable.

Definition 1.4

A $L_{k}^{p}$-map $f:{\bf P}^{n}\rightarrow N$ is said to be (one of ) the standard $S^{1}$-invariant map if $f={\bar{f}}\circ\pi$ where $\pi:{\bf P}^{n}\rightarrow{\bf P}^{n}/S^{1}$ is the quotient map of the standard $S^{1}$-actions on ${\bf P}^{n}$ and ${\bar{f}}:{\bf P}^{n}/S^{1}\rightarrow N$ is the induced map.

Theorem 1.3

Any $v$-stable $L_{k}^{p}$-map on ${\bf P}^{n}$ is stable if it is not (one of ) the standard $S^{1}$-invariant map (up to a conjugation).

Corollary 1.3

For a $v$-stable $L_{k}^{p}$-map, if the identity component $\Gamma_{f}^{0}$ of the isotropy group is nontrivial, it is isomorphic to $S^{1}$. This can happen only when $f$ is one of the standard $S^{1}$-invariant map (up to a conjugation).

After defining the two volume functions in Sec. 2, the theorems stated in this section are proved in Sec. 3.

2 Definitions of the Volume Functions

In this section we give the definitions $v_{m}(f)$ and $v_{m-1}(f)$ of the volume functions.

The definition for $v_{m}(f)$ is standard that we recall now. For Reimannian manifolds $M$ and $N$ and a $C^{1}$ map $f:M\rightarrow N$ with $dim(M)=m$, the $m$-dimensional volume function $v_{m}(f):=vol_{m}(f)=\int_{M}v_{m}(df)d\nu_{M},$ where $d\nu_{M}=|d{\tilde{\nu}_{M}}|$ is the volume density and $d{\tilde{\nu}_{M}}$ is the volume form determined by the metric of $M$. Here $d{\tilde{\nu}_{M}}$ is defined upto a sign and $v_{m}(df)=:vol_{m}(df)$ is a non-negative function defined on $M$ as follows. For any $x\in M,$ let $(e_{1},\cdots,e_{m})$ be a orthonormal frame of $T_{x}M$. Then $v_{m}(df)(x)$ is defined to be the $m$-dimensional volume of the parallelepiped spanned by $df(e_{1}),\cdots,df(e_{m})$ in $T_{f(x)}N$ measured by the metric $g_{f(x)}$ on $T_{f(x)}N.$ Note that $v(df)(x)$ is independent of the choices of the orthonormal frames of $T_{x}M$ since $|df(e_{1})\wedge\cdots\wedge df(e_{m})|=|det(a_{i,j})df(e^{\prime}_{1})\wedge\cdots\wedge df(e^{\prime}_{m})|$ where $e_{i}=\Sigma_{j}a_{i,j}e^{\prime}_{j}$.

In the case that $f$ is a immersion at $x$, hence a local embedding on a small neighborhood $U$ of $x$ with the $m$-dimensional image ${\tilde{U}}$, consider the restriction map $f:U\rightarrow{\tilde{U}}\subset N$ and let $d{\tilde{\nu}}_{m,\tilde{U}}$ be the $m$-dimensional volume form (defined upto a sign) with respect to the induced metric. On $U$, define ${\tilde{v}}_{m}(df)=f^{*}(d{\tilde{\nu}}_{m,\tilde{U}})$. Then $v_{m}(df)=|{\tilde{v}}_{m}(df)/d{\tilde{\nu}}_{m,U}|$.

This implies the next lemma.

Lemma 2.1

Let $u:M_{1}\rightarrow M$ be a diffeomorphism and $f:M\rightarrow N$ be a $C^{1}$-map. Then in a small neighborhood of any $x\in M_{1}$ where $f\circ u$ is a immersion, ${\tilde{v}}_{m}(d(f\circ u))=u^{*}[{\tilde{v}}_{m}(d(f))].$

Corollary 2.1

Let $R_{1}$ be a finite region in $M_{1}$ in the above lemma then $v_{m}(f\circ u|_{R_{1}})=v_{m}(f|_{u(R_{1})})$.

Proof:

Let $M^{*}$ be the open subset of $M$ such that at any point $x\in M^{*}$ $f:M\rightarrow N$ is a immersion and $M_{1}^{*}=u^{-1}(M^{*})$ be the corresponding open set in $M_{1}$. Clearly $v_{m}(f|_{M^{*}})=v_{m}(f)$ and $v_{m}(f\circ u^{-1}|_{M^{*}_{1}})=v_{m}(f\circ u^{-1})$ by the definition. Hence we may assume that $R_{1}$ is lying inside $M^{*}_{1}$ and $u(R_{1})$ is lying inside $M^{*}.$

By using a partition of unit, we only need to consider the local case where we may assume that everything involved is oriented and orientation preserving. Then $v_{m}(df)d\nu_{M}={\tilde{v}}_{m}(df).$

By the lemma,

[TABLE]

[TABLE]

$\Box$

Definition 2.1

A $L_{k}^{p}$-map $f:M\rightarrow N$ is said to be $v_{m}$-stable if $v_{m}(f)>0$.

The $(m-1)$-volume function $v_{m-1}(f)$ is only defined for $f:M\rightarrow N$ that is not $v_{m}$-stable. For such an $f$, $v_{m}(df)(x)=0$ or equivalently $rk(df_{x})<m$ for any $x\in M$.

In this situation, let $M_{m-1}^{*}(f)$ be the open subset of $M$ consisting of points $x$ where $rk(df_{x})=m-1$. If $M_{m-1}^{*}(f)$ is empty, define $v_{m-1}(f)=0.$ Otherwise $v_{m-1}(f)$ is defined as follows.

For any point $x\in M_{m-1}^{*}(f)$, by definition the kernel $K_{x}=\ker(df_{x})$ is $1$-dimensional subspace of the tangent space $T_{x}M$. The collection $K$ of $K_{x}$ with $x\in M_{m-1}^{*}(f)$ is an $1$-dimensional distribution on $M_{m-1}^{*}(f)$ and hence is integrable. Thus we get an $1$-dimensional foliation on $M_{m-1}^{*}(f)$. Locally near a given point $x_{0}\in M_{m-1}^{*}(f)$ by choosing a local $C^{0}$-section of $K$ of unit length, we get an ordinary differential equation so the foliation near $x_{0}$ can be “represented” by the integral curves of the differential equation. The differential equation here is canonically defined up to a choice of a local orientation for $K$, hence is globally defined on the double covering of $M_{m-1}^{*}(f)$ even $K$ is not orientable. Let $S(x_{0})$ be a local slice at $x_{0}$ transversal to the foliation with $S(x_{0})\simeq B^{m-1}$ of a small $(m-1)$-dimensional open ball, and $U_{\epsilon}(S(x_{0}))\simeq S(x_{0})\times(-\epsilon,\epsilon)$ be the collection of integral curves with initial values at $t=0$ lying on $S(x_{0})$ and $t$ varying in the sufficiently small interval $(-\epsilon,\epsilon)$. Each such neighborhood $U_{\epsilon}(S(x_{0}))$ will be called a $K$-neighborhood of $x_{0}$. Then $M_{m-1}^{*}(f)$ can be covered by countably many such open $K$-neighborhoods $U_{\epsilon_{i}}(S(x_{i}))$.

Now fix such a $U_{\epsilon}(S(x_{0}))$ and consider the restriction map $f:U_{\epsilon}(S(x_{0}))\rightarrow N$. Since $f$ is constant along any leaf of the foliation, the image of $U_{\epsilon}(S(x_{0}))$ is the same as that for $f:S(x_{0})\rightarrow N$, which is a local embedding when $S(x_{0})$ is sufficiently small. Let ${\tilde{S}}(x_{0})$ be the image of $S(x_{0})\simeq B^{m-1}$ as a $(m-1)$-dimensional submanifold in $N$. Note that for any other local transversal slice $S(x^{\prime}_{0})$ in $U_{\epsilon}(S(x_{0}))$, the integral curves induce an identification $\gamma_{x_{0}^{\prime},x_{0}}:S(x_{0})\rightarrow S(x^{\prime}_{0})$ of class at least $C^{1}$ such that $f|_{S(x^{\prime}_{0})}\circ\gamma_{x_{0}^{\prime},x_{0}}=f|_{S(x_{0})}$. Thus the images ${\tilde{S}}(x_{0})$ and ${\tilde{S}}(x^{\prime}_{0})$ are the same.

For any point $x\in S(x_{0}),$ let $d{\tilde{\nu}}_{m-1,{\tilde{S}}(x_{0})}$ be the $(m-1)$-dimensional volume form (defined up to a sign) with respect to the induced metric.

Definition 2.2

On $S(x_{0})$, define ${\tilde{v}}_{m-1}(df_{S(x_{0})})=f|_{S(x_{0})}^{*}(d{\tilde{\nu}}_{m-1,{\tilde{S}}(x_{0})})$

and $v_{m-1}(df|_{S(x_{0})})=|{\tilde{v}}_{m-1}(df|_{S(x_{0})})/d{\tilde{\nu}}_{m-1,{S}(x_{0})}|$.

Then the next lemma follows from the definitions above.

Lemma 2.2

Let $u:M_{1}\rightarrow M$ be a diffeomorphism and $f:M\rightarrow N$ as above. Consider a small transversal slice $S(y_{0})$ in $(M^{*}_{1})_{m-1}(f\circ u)$. Let $S(x_{0})=u(S(y_{0}))$ be the corresponding transversal slice in $M^{*}_{m-1}(f)$. Then ${\tilde{v}}_{m-1}(d(f\circ u|_{S(y_{0})}))=u^{*}[{\tilde{v}}_{m-1}(d(f|_{S(x_{0})}))].$

For any two small local slice $S(x_{0})$ and $S(x^{\prime}_{0})$ in $U_{\epsilon}(S(x_{0}))$, ${\tilde{v}}_{m-1}(d(f|_{S(x_{0})}))=\gamma_{x_{0}^{\prime},x_{0}}^{*}[{\tilde{v}}_{m-1}(d(f|_{S(x^{\prime}_{0})}))].$

Definition 2.3

The local $(m-1)$-volume of $f$ over $U_{\epsilon}(S(x_{0}))$ is defined to be $v_{m-1}(f|_{U_{\epsilon}(S(x_{0}))})=\int_{S(x_{0})}{v}_{m-1}(d(f|_{S(x_{0})}))d{\nu}_{S(x_{0})}$, where $d{\nu}_{S(x_{0})}$ is the volume density on $S(x_{0})$.

Then by above lemma, we have

Corollary 2.2

The local $(m-1)$-volume $v_{m-1}(f|_{U_{\epsilon}(S(x_{0}))})$ is well-defined independent of the choice of the local transversal slice. Let $u:M_{1}\rightarrow M$ be a diffeomorphism and $f:M\rightarrow N$ as above lemma. Consider a small $K$-neighborhood $U(S(y_{0}))$ in $(M^{*}_{1})_{m-1}(f\circ u)$. Let $U(S(x_{0}))=u(U(S(y_{0})))$ be the corresponding $K$-neighborhood $U(S(y_{0}))$ in $M^{*}_{m-1}(f)$. Then ${v}_{m-1}(f\circ u|_{U(S(y_{0}))})=v_{m-1}(f|_{U(S(x_{0}))}).$

The proof of the corollary is essentially the same as the proof of the Corollary 2.1. We leave it to the readers.

To define the global $(m-1)$-volume $v_{m-1}(f)$, we define

$v_{m-1}(f|_{\cup_{i=1}^{l}U(S(x_{i}))})$ for the finite union of $K$-sets $\cup_{i=1}^{l}U(S(x_{i}))$ first. The image $\cup_{i=1}^{l}{\tilde{S}}(x_{i}))$ of $\cup_{i=1}^{l}U(S(x_{i}))$ under $f$ is an immersed $(m-1)$ dimensional submanifold with self-intersections. To define $v_{m-1}(f|_{\cup_{i=1}^{l}U(S(x_{i}))})$, we construct partitions of unit on subsets of the image $\cup_{i=1}^{l}{\tilde{S}}(x_{i}))$ first.

Let ${{\beta}}_{i}$ be a smooth bump-off function supported in $U(S(x_{i}))$ and $\beta_{S,i}$ be the restriction of $\beta_{i}$ to $S(x_{i})$. Then $\beta_{S,i}$ can be consiered as a function define on the image ${\tilde{S}}(x_{i})$. Consider the open subset $S^{\prime}(x_{i})$ of $S(x_{i})$ consisting of points $x$ where ${{\beta}}_{i}(x)>0$. Then on the finite union $\cup_{i=1}^{l}{\tilde{S}^{\prime}}(x_{i}))$ of the images of $S^{\prime}(x_{i}),$ the nonzero bump-off functions $\beta_{S,i},i=1,\cdots,l$ define a partition of unit in the usual manner. Denote the bump-off function of the partition of unit on ${\tilde{S}^{\prime}}(x_{i})\simeq S^{\prime}(x_{i})$ by $\alpha_{i}$.

For each choice of $\beta$ above, define

[TABLE]

[TABLE]

Definition 2.4

The $(m-1)$-volume functions are defined by:

(1) $v_{m-1}(f|_{\cup_{i=1}^{l}U(S(x_{i}))})=\sup_{\beta}v_{m-1,\beta}(f|_{\cup_{i=1}^{l}U(S(x_{i}))})$ over all bump-off functions $\beta$; (2) $v_{m-1}(f)$ is equal to the supremum of $v_{m-1}(f|_{\cup_{i=1}^{l}U(S(x_{i}))})$ over all finite union of $K$-subsets $\cup_{i=1}^{l}U(S(x_{i}))$ in $M^{*}_{m-1}.$

For the proofs in next section only the properties for the local volume $v_{m-1}$ stated in above lemma are used. The definition above for the global $(m-1)$-volume is to ensure that if $v_{m-1}(f)>0$, so is $v_{m-1}(f|_{U(S(x_{0}))})>0$ for some $K$-set $U(S(x_{0}))$ in $M^{*}_{m-1}.$

The functions $v_{i}(f)$ for all $0<i\leq m=\dim(M)$ can be defined using similar ideas. However for the proofs in this paper only above two functions are useful.

3 Proof of the Main Theorems

3.1 Proof of Theorem 1.1, Proposition 1.1, Corollary 1.1 and Corollary 1.2

We make a reduction first.

Lemma 3.1

A group action $\Phi:G\times X\rightarrow X$ is proper if and only if for any point $p\in X\times X$ there is a neighborhood $U$ of $p$ in $X\times X$ such that the closure of $\pi_{G}((\Phi\times{\it id}_{X})^{-1}(U))$ is compact in $G$.

The proof of the lemma is elementary and is given in [Liu 2015].

By the above lemma, the properness of the action of $G$ on ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ can be derived from the following theorem.

Theorem 3.1

The action of $G={PSL(n+1,{\bf C})}$ on ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ has the following property: for any $f_{1}$ and $f_{2}$ there exist the open neighborhoods $U_{\epsilon_{1}}(f_{1})$ and $U_{\epsilon_{2}}(f_{2})$ containing $f_{1}$ and $f_{2}$ and compact subsets $K_{1}$ and $K_{2}$ in $G$ accordingly such that for any $h_{1}$ in $U_{\epsilon_{1}}(f_{1})$ ($h_{2}$ in $U_{\epsilon_{2}}(f_{2})$) and $g_{1}$ in $G\setminus K_{1}$ ($g_{2}$ in $G\setminus K_{2}$ ), $g_{1}\cdot h_{1}$ is not in $U_{\epsilon_{2}}(f_{2})$ ($g_{2}\cdot h_{2}$ is not in $U_{\epsilon_{1}}(f_{1})$).

Proof:

We start with some elementary linear algebra. For any $g\in SL(n+1,{\bf C}),$ we have a decomposition in $SL(n+1,{\bf C}),$ $g=h\cdot u$ with $u\in SU(n+1)$ and $h$ being self-adjoint and positive. Indeed $h=(g\cdot g^{*})^{\frac{1}{2}}\in SL(n+1,{\bf C})$ and $u=(g\cdot g^{*})^{-\frac{1}{2}}\cdot g\in SU(n+1)$. Consider the decomposition $h=w^{*}\cdot diag(r_{1},r_{2}\cdots,r_{n+1})\cdot w$. Then $g=w^{*}\cdot diag(r_{1},r_{2}\cdots,r_{n+1})\cdot wu.$ Here $r_{i}>0$ for $i=1,\cdots,n+1.$ Rename $w^{*}$ as $u$ and $wu$ as $v$. Denote $diag(r_{1},r_{2}\cdots,r_{n+1})$ by $\Delta({\bf r})$ for short. Then we have the decomposition $g=u\cdot\Delta({\bf r})\cdot v$ in $SL(n+1)$ with $u$ and $v$ in $SU(n+1)$. This decomposition is not unique for a non generic $g$, but we only need the existence of the decomposition.

We always assume that ${\bf r}$ is ordered as $0<r_{1}\leq r_{2}\leq\cdots\leq r_{n+1}.$ Then as an element in $PSL(n+1)$, we may assume that $r_{n+1}=1$ and $\Delta({\bf r})=diag(r_{1},r_{2}\cdots,r_{n},1).$ Denote the smallest element $r_{1}$ by $a$ and $\Delta({\bf r})$ by $\Delta(a)$.

Assume that the Theorem 2.1 (a) is not true. Then for any neighbourhoods $U_{\epsilon_{i}}(f_{i}),i=1,2$ and any nested sequences of compact sets $K_{1}\subset K_{2}\subset\cdots\subset K_{l}\cdots$ in $G$, there are sequences $\{g_{k}\}_{k=1}^{\infty}$ in $G$ and $\{h_{k}\}_{k=1}^{\infty}$ in $U_{\epsilon_{1}}(f_{1})$ such that (a) $g_{k}$ is not in $K_{i(k)}$; (b) $h_{k}\circ g_{k}$ is in $U_{\epsilon_{2}}(f_{2}).$

Here $U_{\epsilon_{1}}(f_{1})$, $U_{\epsilon_{2}}(f_{2})$ and $K_{k},k=1,\cdots$, will be decided below. Note that we allow $f_{1}=f_{2}$ but the choice of $U_{\epsilon_{1}}(f_{1})$ and $U_{\epsilon_{2}}(f_{2})$ below are different.

Let $D_{n+1}$ be the collection of all non-singular diagonal matrices with $n+1$ positive entries. Choose ${\tilde{K}}_{k}\subset SU(n+1)\times D_{n+1}\times SU(n+1)$ to be $\{(u,\Delta({\bf r}),v)\in SU(n+1)\times D_{n+1}\times SU(n+1)\,|\,\frac{1}{n}\leq r_{i}\leq 1,i=1,\cdots,k+1\}$, where $\Delta({\bf r})=diag(r_{1},\cdots,r_{n+1})$. Denote the corresponding compact set in $PSL(n+1)$ by ${K}_{n}$ obtained by sending $(u,{\bf r},v)$ to $u\cdot\Delta({\bf r})\cdot v$.

First fix $U_{\epsilon_{1}}(f_{1})$ without any restrictions.

Now we need to deal with the two case: (I) $v_{m}(f_{2})>0$ and (II) $v_{m}(f_{2})=0$ but $v_{m-1}(f_{2})>0$. Here and below $m=2n$.

For case (I) we may assume that $v_{m}(f_{1})>0$ as well. Indeed if in this case $v_{m}(f_{1})=0$ but $v_{m-1}(f_{1})>0$, by replacing $g_{k}$ by $g^{-1}_{k}$ and switching the roles of $f_{1}$ and $f_{2}$, it is reduced to one of the sub-cases of case (II). Though it is possible to give an unified proof for both cases together, in the following we give proofs for each of the cases.

$\bullet$ Proof for case (I).

In this case, we may assume that $v_{m}(h)>0$ for all the $L_{k}^{p}$-maps $h$ involved so that only the function $v_{m}$ is used.

Now we choose $\epsilon_{2}$ for $U_{\epsilon_{2}}(f_{2})$ as follows. Since $f_{2}$ is $v_{m}$-stable, its volume $v_{m}(f_{2})=\delta_{2}>0.$ Then there is a point $x_{0}\in{\bf CP}^{n}$ such that $v_{m}(df_{2})(x_{0})>0.$ Hence there are positive constants $\gamma$ and $\rho$ small enough such that for any $x$ in the ball $B(x_{0};\rho)$ of radius $\rho$ centered at $x_{0}$, $v_{m}(df_{2})(x)>\gamma$. Then there is an ${\tilde{\epsilon}}_{2}>0$ such that for any $h$ with $\|h-f_{2}\|_{C^{1}}<{\tilde{\epsilon}}_{2}$, the same is true.

Now $\|h-f_{2}\|_{C^{1}}\leq C_{2}\cdot\|h-f_{2}\|_{k,p}$ by our assumption. Hence we choose $\epsilon_{2}$ by the requirement that $\epsilon_{2}<{\tilde{\epsilon}}_{2}/C_{2}$ so that for any $h\in U_{\epsilon_{2}}(f_{2})$, $\|h-f_{2}\|_{C^{1}}<{\tilde{\epsilon}}_{2}$.

With this choice of $\epsilon_{2}$, for any $h\in U_{\epsilon_{2}}(f_{2})$, and any point $x\in B(x_{0};\rho)$, $v_{m}(dh)(x)>\gamma$.

In these notations, the condition (a) above implies that for $g_{k}$ in $G$ with $g_{k}=u_{k}\cdot D({a}_{k})\cdot v_{k}$, we have $lim_{k\mapsto\infty}a_{k}=0.$

After taking subsequence, we may assume that $lim_{k\mapsto\infty}u_{k}=u$ and $lim_{k\mapsto\infty}v_{k}=v$ in $SU(n+1)$

Note that when considered as automorphisms on ${\bf CP}^{n}$, the convergence here are with respect to $C^{\infty}$-topology on the corresponding mapping space.

Let ${\bf CP}^{n}={\bf C}^{n}\cup{\bf CP}^{n-1}$. Now we have the following two cases: (A) ${\bf CP}^{n-1}\cap v(B(x_{0};\rho)=\varphi$ and (B) ${\bf CP}^{n-1}\cap v(B(x_{0};\rho)\not=\varphi.$ Since $v$ is an isometry, it is easy to see that in both cases, there exits an $x_{1}\in B(x_{0};\rho)$ and a positive number $\rho_{1}<<\rho$ such that $v(B(x_{1};\rho_{1}))\cap{\bf CP}^{n-1}=\varphi.$ We may assume that $dist(v(B(x_{1};\rho_{1})),{\bf CP}^{n-1})>\delta>0$. Then for $i>i_{0}$ large enough, $dist(v_{i}(B(x_{1};\rho_{1})),{\bf CP}^{n-1})>\delta$ as well. Hence there is a large $R$ such that for $i>i_{0}$ large enough, $v_{i}(B(x_{1};\rho_{1}))$ is lying inside $D^{n}_{R}=:D(R_{1})\times\cdots\times D(R_{n})\subset{\bf C}^{n}$ with $R_{1}=R_{2}\cdots=R_{n}=R$ of the $n$-fold product of the open disks centered at origin in ${\bf C}$ with radius $R$.

Note that in term of the coordinate of ${\bf C}^{n}\subset{\bf CP}^{n}$, the action of $\Delta({a_{i}})$ is given by $\Delta({a_{i}})(z)=(a_{i}\cdot z_{1},r_{i,2}\cdot z_{2},\cdots,r_{i,n}\cdot z_{n})$ with $a_{i}\leq r_{i,2}\cdots\leq r_{i,n}\leq 1.$

Hence for any fixed $R>0$ and any given $\epsilon>0,$ our assumption that $a_{i}\mapsto 0$ implies that there is a fixed $i_{0}(\epsilon)>>0$ such that when $i>i_{0}(\epsilon),$ $\Delta({a_{i}})(v_{i}(B(x_{1};\rho_{1})))\subset a_{i}\cdot D(R_{1})\times r_{i,2}\cdot D(R_{2})\times\cdots\times r_{i,n}\cdot D(R_{n})\subset D(\epsilon)\times D(R_{2})\times\cdots\times D(R_{n})$ with $R_{k}=R$ for $k=1,\cdots,n.$

Hence the $vol(\Delta({a_{i}})(v_{i}((B(x_{1};\rho_{1}))))$ $\leq C_{3}\epsilon^{2}R^{2n-2}$. Here the volumes are computed with respect to the Fubini-Study metric which is uniformly equivalent to the flat metric on $D^{n}_{R}$ for fixed $R$. Applying this to $g_{i}=u_{i}\circ\Delta({a_{i}})\circ v_{i}$, since $u_{i}$ preserves the Fubini-Study metric, we conclude that for $i$ large enough, $vol(g_{i}(B(x_{1};\rho_{1})<C_{4}\epsilon^{2}.$

Now

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By letting $\epsilon\mapsto 0,$ we conclude that $\lim_{i\mapsto\infty}v_{m}((h_{i}\circ g_{i})|_{B(x_{1};\rho_{1})})=0.$

Now $h_{i}\circ g_{i}\in U_{\epsilon_{2}}(f_{2})$. Recall that for any $h\in U_{\epsilon_{2}}(f_{2})$, $v_{m}(h|_{B(x_{1};\rho_{1})})>C\cdot\gamma\cdot\rho^{2n}_{1}$, which is a fixed positive constant. This is a contradiction.

$\Box$

$\bullet$ Proof for case (II).

The proof for this case is a modification of the proof above for case (I). We still argue by contradiction. The choices for $\epsilon_{1}$ and $K_{k}$ are the same as above.

Now we choose $\epsilon_{2}$ for $U_{\epsilon_{2}}(f_{2})$ as follows. Recall in this case, we may assume that $v_{m}(f_{2})=0$ but $v_{m-1}(f_{2})>0.$ Assume that $v_{m-1}(f_{2})=\delta_{2}>0.$ Then there is a point $x_{0}\in{\bf CP}^{n}$ such that $v_{m-1}(df_{2}|_{U})(x_{0})>0$ for some small open $K$-subset $U$. Hence there are positive constants $\gamma$ and $\rho$ small enough such that (1) there is a $(m-1)$-dimensional geodesic ball $B_{m-1}(x_{0};\rho)$ of radius $\rho$ centered at $x_{0}$ and (2) for any $x$ in $B_{m-1}(x_{0};\rho)$, the $v_{m-1}(df_{2}|_{B_{m-1}(x_{0};\rho)})(x)>\gamma$. Then there is an ${\tilde{\epsilon}}_{2}>0$ such that for any $h$ with $\|h-f_{2}\|_{C^{1}}<{\tilde{\epsilon}}_{2}$, the same is true: $v_{m-1}(dh|_{B_{m-1}(x_{0};\rho)})(x)>\gamma$.

As before, since $\|h-f_{2}\|_{C^{1}}\leq C_{2}\cdot\|h-f_{2}\|_{k,p}$ by choosing $\epsilon_{2}$ such that $\epsilon_{2}<{\tilde{\epsilon}}_{2}/C_{2}$, we have that for any $h\in U_{\epsilon_{2}}(f_{2})$, $\|h-f_{2}\|_{C^{1}}<{\tilde{\epsilon}}_{2}$.

With this choice of $\epsilon_{2}$, for any $h\in U_{\epsilon_{2}}(f_{2})$, and any point $x\in B_{m-1}(x_{0};\rho)$, $v_{m-1}(dh|_{B_{m-1}(x_{0};\rho)})(x)>\gamma$.

Let ${\bf CP}^{n}={\bf C}^{n}\cup{\bf CP}^{n-1}$. Since $m-1=2n-1>\dim({\bf CP}^{n-1})$, as before, we may assume that there exits an $x_{1}\in B_{m-1}(x_{0};\rho)$ and a positive number $\rho_{1}<<\rho$ such that $v(B_{m-1}(x_{1};\rho_{1}))\cap{\bf CP}^{n-1}=\varphi$ where $B_{m-1}(x_{1};\rho_{1})$ is the ($(m-1)$-dimensional) ball of radius $\rho_{1}$ centered at $x_{1}$ inside $B_{m-1}(x_{0};\rho)$. Then $dist(v(B_{m-1}(x_{1};\rho_{1})),{\bf CP}^{n-1})>\delta>0$ and $dist(v_{i}(B_{m-1}(x_{1};\rho_{1})),{\bf CP}^{n-1})>\delta$ for $i$ large enough.

Hence there is a large $R$ such that for $i>i_{0}$ large enough, $v_{i}(B_{m-1}(x_{1};\rho))$ is lying inside $D^{n}_{R}=:D(R_{1})\times\cdots\times D(R_{n})\subset{\bf C}^{n}\simeq{\bf R}^{2n}$ of the $n$-fold product of the open disks centered at origin in ${\bf C}$ with radius $R_{i}=R$.

Now the key point is to show that $vol_{m-1}[\Delta({a_{i}})(v_{i}(B_{m-1}(x_{1};\rho_{1})))]$ tends to zero as $i$ goes to infinity. By projecting to one of the $(2n-1)$-dimensional coordinate planes, the tangent planes of $v(B_{m-1}(x_{1};\rho_{1})))$ and $v_{i}(B_{m-1}(x_{1};\rho_{1})))$ at $x_{1}$ can be realized a graph of a linear function. Hence for $\rho_{1}$ small enough, $v(B_{m-1}(x_{1};\rho_{1})))$ and $v_{i}(B_{m-1}(x_{1};\rho_{1})))$ can be realized a graph of a function as well over the coordinate plane with the dimension $2n-1=m-1$. Hence at least one of the first two coordinate lines in ${\bf R}^{2n}\simeq{\bf C}^{n}$ has to be included in the $(2n-1)$-dimensional coordinate plane.

Now recall that in term of the coordinate of ${\bf C}^{n}\subset{\bf CP}^{n}$, the action of $\Delta({a_{i}})$ is given by $\Delta({a_{i}})(z)=(a_{i}\cdot z_{1},r_{i,2}\cdot z_{2},\cdots,r_{i,n}\cdot z_{n})$ with $a_{i}\leq r_{i,2}\cdots\leq r_{i,n}\leq 1.$ By renaming the coordinate line transversal to the $(2n-1)$-dimensional coordinate plane as the last coordinate line of ${\bf R}^{2n}$, we may assume that the $(2n-1)$-dimensional coordinate plane are given by the coordinates $(x_{1},\cdots,x_{2n-1})$. The action $\Delta({a_{i}})$ then has the form $\Delta({a_{i}})(x_{1},\cdots,x_{2n-1},x_{2n})=(a_{i}\cdot x_{1},r_{i,2}\cdot x_{2},\cdots,r_{i,K-1}\cdot x_{K-1},r_{i,K}\cdot x_{K},\cdots,r_{i,2n}\cdot x_{2n})$ with the property that (1) $a_{i}\leq r_{i,1}\cdots\leq r_{i,2n-1}\leq 1;$ (2) there is a $K\geq 2$ such that $\lim_{i\mapsto\infty}r_{i,j}=0$ for $j<K$ and $r_{i,j}$ is bounded below from zero for $K\leq j\leq 2n-1;$ (3) $r_{i,2n}\leq 1$. Then $\Delta({a_{i}})$ is decomposed as $\Delta({a_{i}})=\Delta^{<K}({a_{i}})\circ\Delta^{\geq K}({a_{i}})$. Here the action of $\Delta^{<K}({a_{i}})$ is the same as that $\Delta({a_{i}})$ on the first $(K-1)$ variables but the identity on the rest of variables; and $\Delta^{\geq K}({a_{i}})$ is just another way around.

Now using the family of $(2n-(K-1))$-dimensional planes parallel to the coordinate plane of the last $(2n-(K-1))$ coordinates of ${\bf R}^{2n}$ to slice $v(B_{m-1}(x_{1};\rho_{1})))$ and $v_{i}(B_{m-1}(x_{1};\rho_{1}))),$ they are realized as families of $(2n-K)$-dimensional balls (generically) of bounded volumes over the $(K-1)$-dimensional finite ellipsoids $E$ and $E_{i}$ in the coordinate plane of the first $(K-1)$ coordinates. Denote these families by $\pi:v(B_{m-1}(x_{1};\rho_{1})))\rightarrow E$ and $\pi_{i}:v_{i}(B_{m-1}(x_{1};\rho_{1})))\rightarrow E_{i}.$ Let $M$ be an up bound of $(2n-K)$-dimensional volume of the balls (fibers ) in the family $\pi$. Since $v_{i}\mapsto v$ in $C^{\infty}$-topology, we may assume that $M$ is also an up bound of the $(2n-K)$-dimensional volumes of the fibers of the family $\pi_{i}$ for $i$ large enough. Then by Fubini’s theorem, $vol_{m-1}(v_{i}(B_{m-1}(x_{1};\rho_{1})))\leq M\cdot vol_{K-1}(E_{i})$. Since $\Delta^{\geq K}({a_{i}})$ acts as identity in the first $K-1$ variables with a dilation factor less than $1$ on the rest of variables, we still have

[TABLE]

Now applying $\Delta^{<K}({a_{i}})$ by noting that $\Delta^{<K}({a_{i}})$ only acts on the first $K-1$ variables leaving the rest unchanged,

[TABLE]

[TABLE]

[TABLE]

Now $E$ and hence $E_{i}$ are lying in a finite ball of a large radius $R$ in the coordinate plane of the first $K-1$ variables.

By the definition, $vol_{K-1}(\Delta^{<K}({a_{i}})(E_{i}))\leq r_{i,K-1}vol_{K-1}(E_{i}))\leq C\cdot r_{i,K-1}R^{K-1}$ that goes to zero as $i$ goes to infinity. Hence $vol_{m-1}(\Delta(a_{i})(v_{i}(B_{m-1}(x_{1};\rho_{1}))))$ $\mapsto 0$ as $i\mapsto\infty$. Note that in above the volume is computed in Euclidean metric. Since $v_{i}(B_{m-1}(x_{1};\rho_{1}))$ and $\Delta(a_{i})(v_{i}(B_{m-1}(x_{1};\rho_{1})))$ are lying in a bounded region $D_{R}^{n}\subset{\bf C}^{n}$, the same conclusion is true with respect to the Fubini-Study metric. Then as before, we conclude from above that $vol_{m-1}(g_{i}(B_{m-1}(x_{1};\rho_{1})\mapsto 0$ as $i\mapsto\infty$.

The rest of the proof is the same as the one for case I by replacing $B(x_{1};\rho_{1})$ there by $B_{m-1}(x_{1};\rho_{1})$ and $m$-volumes there by the corresponding $(m-1)$-volumes.

Indeed

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We conclude that $\lim_{i\mapsto\infty}v_{m-1}((h_{i}\circ g_{i})|_{B_{m-1}(x_{1};\rho_{1})})=0.$

Now $h_{i}\circ g_{i}\in U_{\epsilon_{2}}(f_{2})$. Recall that for any $h\in U_{\epsilon_{2}}(f_{2})$, $v_{m-1}(h|_{B(x_{1};\rho_{1})})>C\cdot\gamma\cdot\rho^{2n-1}_{1}$, which is a fixed positive constant. This is a contradiction.

$\Box$

Corollary 3.1

Corollary 1.1 holds.

Proof:

We need to show that $\Gamma_{f}$ is compact if $f$ is $v$-stable.

By taking $f=f_{1}=f_{2}$ with $f$ being $v$-stable, the above theorem implies that there is a compact subset $K\subset G$ such that $\Gamma_{f}$ is contained in $K$. It is well-known that the action map $\Phi:G\times{{\cal M}}_{k,p}(M,N)\rightarrow{{\cal M}}_{k,p}(M,N)$ is continuous (for a proof, see [Liu 2017] for instance ) so that $\Gamma_{f}$ is closed.

$\Box$

Corollary 3.2

The Corollary 1.2 holds.

Proof:

We need to prove that the $G$-orbit of $f$ in ${{\cal M}}^{*}_{k,p}$ is closed.

Rename $f$ as $f_{1}$. If the corollary is not true, there exist $g_{i}\in G$ and $f_{2}\in{{\cal M}}^{*}_{k,p}$ such that $f_{2}=\lim_{i\mapsto\infty}f_{1}\circ g_{i},$ but $f_{2}$ is not in $G\cdot f_{1}$. Therefore for any $U_{\epsilon_{2}}(f_{2}),$ when $i$ is large enough, $f_{1}\circ g_{i}$ is in $U_{\epsilon_{2}}(f_{2}).$ On the other hand, the Theorem 3.1 with the same notation implies that for all such $i$, $g_{i}$ is in the compact set $K_{1}$. Therefore, we may assume that $\lim_{i\mapsto\infty}g_{i}=g$ in $K_{1}.$ Consequently, $f_{2}=\lim_{i\mapsto\infty}f_{1}\circ g_{i}=f_{1}\circ g.$ That is $f_{2}\in G\cdot f_{1}$ which is a contradiction.

$\Box$

We restate the Propostion 1.1 below.

Proposition 3.1

Let $G=PSL(n+1,{\bf C})$. Then the space ${{\cal M}}^{*}_{k,p}({\bf P}^{n},N)$ of $v$-stable $L_{k}^{p}$-maps on ${\bf P}^{n}$ is $G$-Hausdorff in the sense that for any two diffent $G$-orbits $Gf_{1}$ and $Gf_{2}$, there exit $G$-neighborhoods $GU_{1}$ and $GU_{2}$ such that $GU_{1}\cap GU_{2}=\varphi$. Therefore, the quotient space ${{\cal M}}_{k,p}({\bf P}^{n},N)/G$ of unparametrized $v$-stable $L_{k}^{p}$-maps is Hausdroff.

Proof:

By Theorem 3.1, for any $g$ not in the compact set $K_{1}$ and $h\in U_{\epsilon_{1}}(f_{1})$, $h\circ g$ is not in $U_{\epsilon_{2}}(f_{2})$. By our assumption, we may assume that $U_{\epsilon_{1}}(f_{1})$ and $U_{\epsilon_{2}}(f_{2})$ have no intersection.

$\bullet$ Claim: when $\epsilon_{i},i=1,2$ are small enough, $(G\cdot U_{\epsilon_{1}}(f_{1}))\cap U_{\epsilon_{2}}(f_{2})$ is empty.

Proof:

If this is not true, there are $h_{i}\in U_{\delta_{i}}(f_{1})$ and $g_{i}\in K_{1}$ such that $h_{i}\circ g_{i}$ is in $U_{\delta_{i}}(f_{2})$ with $\delta_{i}\mapsto 0.$ The compactness of $K_{1}$ implies that after taking a subsequence, we have that $\lim_{i\mapsto\infty}g_{i}=g\in K_{1}.$ Since $\delta_{i}\mapsto 0$, we have that $f_{1}=\lim_{i\mapsto\infty}h_{i}$ and $f_{2}=\lim_{i\mapsto\infty}h_{i}\circ g_{i}=f_{1}\circ g.$ Hence, $f_{1}$ and $f_{2}$ are in the same orbit which contradicts to our assumption. Note that in the last identity above, we have used the fact that the action map $\Psi:G\times{{\cal M}}_{k,p}({\bf P}^{n},N)\rightarrow{{\cal M}}_{k,p}({\bf P}^{n},N)$ is continuous.

$\Box$

Of course the same proof also implies that $(G\cdot U_{\epsilon_{2}}(f_{2}))\cap U_{\epsilon_{1}}(f_{1})$ is also empty for sufficiently small $\epsilon_{i},i=1,2.$

If $h\in(G\cdot U_{\epsilon_{1}}(f_{1}))\cap(G\cdot U_{\epsilon_{2}}(f_{2})),$ then there are $h_{i}\in U_{\epsilon_{i}}(f_{i})$ and $g_{i}\in G,i=1,2$ such that $h=h_{1}\circ g_{1}=h_{2}\circ g_{2}.$ Hence $h_{2}=h_{1}\circ g_{1}\circ g_{2}^{-1}$ and $(GU_{\epsilon_{1}}(f_{1}))\cap U_{\epsilon_{2}}(f_{2})$ is not empty. This contradicts to the above claim.

$\Box$

3.2 Proof of Theorem 1.2 and Theorem 1.3

We need show that (i) any $v_{2n}$-stable $L_{k}^{p}$-map $f$ with the domain ${\bf CP}^{n}$ is stable; (ii) if $v_{2n}(f)=0$ but $v(f)>0$, then $f$ is either a stable map or one of $S^{1}$-invariant maps (up to an conjugation).

Proof:

Since in both cases $v_{(}f)>0$, ${\Gamma}_{f}$ is compact. If the identity component ${\Gamma}^{0}_{f}$ is nontrivial, let ${\tilde{\Gamma}}^{0}_{f}$ be the lifting of ${\Gamma}^{0}_{f}$ in $SL(n+1,{\bf C})$. Since maximal compact and connected subgroup $SU(n+1)$ has only one orbit inside $SL(n+1,{\bf C})$ under conjugations, we may assume that ${\tilde{\Gamma}}^{0}_{f}$ is contained in $SU(n+1)$ so that ${\Gamma}^{0}_{f}$ is contained in $SU(n+1)/{\bf Z}_{2}^{n+1}$. Thus up to a conjugation, we may assume that ${\tilde{\Gamma}}^{0}_{f}$ contains a subgroup $S^{1}$ inside the maximal tours $T^{n+1}\subset SU(n+1)$ such that the induced action of $S^{1}\subset{\tilde{\Gamma}}^{0}_{f}$ on ${\bf P}^{n}$ given by $e^{\theta}\cdot(z_{0}:z_{1}:\cdots:z_{n})=(e^{m_{0}\theta}z_{0}:e^{m_{1}\theta}z_{1}:\cdots:e^{m_{n}\theta}z_{n})$ is nontrivial. It has the form $e^{\theta}\cdot(z_{1},\cdots,z_{n})=(e^{m_{1}\theta}z_{1},e^{m_{2}\theta}z_{2},\cdots,e^{m_{n}\theta}z_{n})$ with some of $m_{i}\not=0,$ with respect to the coordinate $z=(z_{1},\cdots,z_{n})$ of ${\bf C}^{n}$ for a proper choice of the decomposition ${\bf CP}^{n}={\bf C}^{n}\cup{\bf CP}^{n-1}.$ The $S^{1}$-action here is the so called (one of ) standard $S^{1}$-action in (ii) above.

If $\dim(L(\Gamma_{f}^{0}))\geq 2$, let $\xi_{1}$ and $\xi_{2}$ be two linear independent elements in $L(\Gamma_{f}^{0})=L({\tilde{\Gamma}}^{0}_{f})$ with the corresponding $1$-parameter subgroups $S^{1}_{\xi_{1}}$ and $S^{1}_{\xi_{2}}$ both $\simeq S^{1}$, each acting on ${\bf P}^{n}$ in the form above up to a conjugation. Since both action come from/extend to the corresponding $C^{*}$-actions, if the two actions of $S^{1}_{\xi_{1}}$ and $S^{1}_{\xi_{2}}$ are identical on some open set of ${\bf P}^{n}$, so are the corresponding $C^{*}$-actions. Since the $C^{*}$-actions are algebraic, the two actions of $S^{1}_{\xi_{1}}$ and $S^{1}_{\xi_{2}}$ are the same if they agree on an open set.

Now consider the map from $L(\Gamma_{f}^{0})$ to the set of vector fields ${\cal V}({\bf P}^{n})$ given by sending $\xi$ to the vector field $X_{\xi}$ that generates the action of $S^{1}_{\xi}.$ The special form of the $S^{1}$-action above implies that the map is injective. Let $L(\xi_{1},\xi_{2})$ be the linear span of $\xi_{1}$ and $\xi_{2}$. Then above argument implies that the ”plane” field $L(X_{\xi_{1}},X_{\xi_{2}})$ is $2$-dimensional generically in the sense that the set of points $x$ where $\dim[L(X_{\xi_{1}},X_{\xi_{2}})(x)]<2$ has no interior point. Now by definition $L(X_{\xi_{1}},X_{\xi_{2}})(x)\subset L(\Gamma_{f}^{0})$ is contained in $\ker(df_{x})$. Hence at any generic point $x$, the $rk(df_{x})<m-1$. Hence the same is true for all point $x\in{\bf P}^{n}$ by the lower semi-continuity of the function $x\rightarrow rk(df_{x})$, which contradicts to $v_{m-1}(f)>0.$ Hence $\dim L(\Gamma_{f}^{0})=1$ and $\Gamma_{f}^{0}\simeq S^{1}.$

We have proved that if $v(f)>0$ and ${\Gamma}^{0}_{f}$ is nontrivial, then $f$ is one of the standard $S^{1}$-invariant map up to a conjugation. This proves (ii).

To prove (i), still assume the identity component ${\Gamma}^{0}_{f}$ is nontrivial, hence with the $S^{1}$-action above. Assume that $m_{1}\not=0.$ Since $f$ is $v_{2n}$-stable, $df_{v}$ is injective at some point $x_{0}$ so that $f_{v}$ is a local embedding on a small $\epsilon$-polydisk $D^{n}_{\epsilon}(x_{0})=:D_{\epsilon}(x_{0,1})\times\cdots\times D_{\epsilon}(x_{0,n})\subset{\bf C}^{n}$ where $D_{\epsilon}(x_{0,j})$ is the open disk in ${\bf C}$ centered at $x_{0,j}$ with radius $\epsilon$.

By taking a smaller $\epsilon$-polydisk, we may assume that the origin $0\in{\bf C}$, which is the fixed point of the nontrivial action $\phi_{i}(\theta,z)=e^{m_{i}\theta}z$ is not contained in $D_{\epsilon}(x_{0,i})$.

Then for $\phi\not={\it id}\in S^{1}$ but sufficiently close to ${\it id}$, the image $\phi(D_{\epsilon/2}(x_{0,i}))$ is different from $D_{\epsilon/2}(x_{0,i})$ but still inside $D_{\epsilon}(x_{0,i})$ if $\phi_{i}$ is nontrivial, where $\phi_{i}$ is the $i$-th component of $\phi$ so that the image $\phi_{i}(D^{n}_{\epsilon/2}(x_{0}))$ is different from $D^{n}_{\epsilon/2}(x_{0}).$ Since $f$ is an embedding on $D^{n}_{\epsilon}(x_{0})$, the images $f(D^{n}_{\epsilon/2}(x_{0}))$ and $f(\phi(D^{n}_{\epsilon/2}(x_{0})))=f\circ\phi(D^{n}_{\epsilon/2}(x_{0}))$ then are different. Hence on $D^{n}_{\epsilon/2}(x_{0})$, $f\circ\phi\not=f$ already so that $\phi$ is not lying inside the isotropy group of $f$. This is a contradiction.

$\Box$