中心仿射超曲面的若干变分公式

Some Variational Formulae of the Centro-Affine Hypersurfaces

摘要: 对于仿射空间中的非退化超曲面，运用中心仿射几何量在自然参数下的表示，给出了中心仿射体积的第一和第二变分公式简单的直接证明。进一步研究了Techebychev形式长度平方积分以及3形式长度平方积分的变分公式。

Abstract: The centro-affine differential geometric invariants of a piece of non-degenerate hypersurface in an affine space are represented with respect to the natural parametrization of the given hypersurface. Then a simple and direct proof of the first and the second variation formulae of the centro-affine volume is presented. Furthermore, the variational formulae of the integrals of the square of the length of the Tchebychev field and the cubic form are investigated.

1. 引言

仿射空间中非退化超曲面的中心仿射几何是超曲面相对微分几何的一个重要分支，关于其一般理论介绍可参考文献 [1] [2] [3] 。近来这种超曲面几何在Finsler几何中有重要的应用，可参考文献 [4] [5] 。设 $f:D\to R^{n+1}$ 是嵌入映射，其中D是 $R^n$ 中的连通开域。那么 $M=f\left(D\right)$ 称为 $R^{n+1}$ 中嵌入超曲面。本文假设M非退化，且其上有中心法化诱导的 $GL\left(n+1;R\right)$ 不变黎曼度量 $g$。王长平在 [6] 中得到了M的仿射体积 $V={\displaystyle {\int }_D\text{d}{V}_M}={\displaystyle {\int }_D\sqrt{\det \left(g_{ij}\right)}}\text{d}{x}^1\wedge \cdots \wedge \text{d}{x}^n$ 的变分公式。

定理1 [6] . 设 $f:D\times \left(-\epsilon ,\epsilon \right)\to R^{n+1}$ 是超曲面M的正则变分，其变分向量场在自然标架场下，可表示为 $f^{\prime }=\frac{\partial f}{\partial t}=\phi f+\psi$，其中 $\phi$ 是D上的光滑函数， $\psi$ 是切向分量，并且二者均D在的边界为零。那么M的体积第一变分公式为

$V^{\prime }\left(t\right)=-\frac{n}{2}{\displaystyle {\int }_D\phi H}\text{d}{V}_{M_t}$

对于满足 $H=0$ 超曲面，有下面的体积第二变分公式

$V{\left(0\right)}^{\prime \text{}\prime }=-\frac{1}{4}{\displaystyle {\int }_D\left[\Delta \phi \left[\Delta \phi +2\left(n+1\right)\phi \right]-{\left(nT\left(\nabla \phi \right)\right)}^2+2nC\left(\nabla \phi ,\nabla \phi \right)\left(T\right)\right]\text{d}{V}_M}$

其中 $H=d^{*}T=T_{,k}^k$ 称为 $M$ 超曲面的中心仿射平均曲率， $T=\frac{1}{n}trC$ 为Techebychev形式， $C$ 为3形式。

满足 $H=0$ 超曲面称为中心仿射极小曲面，这类曲面的相关研究可参考 [6] [7] [8] [9] ，与之密切相关的Chebychev超曲面的研究可参考 [10] [11] [12] 。

定理1最先由王长平在文献 [6] 中运用活动标架法得到，但证明比较复杂。本文采用超曲面的参数表示，给出上述变分公式的简单初等证明。运用这种方法，我们进一步计算超曲面的另外两个仿射不变量的变分公式。

定理2. 对于变分向量场为 $f^{\prime }=\phi f$ 的正则变分，泛函 $\mathfrak{T}={\displaystyle {\int }_D{\Vert T\Vert }^2\text{d}{V}_M}$ 的一阶变分为

${\mathfrak{T}}^{\prime }\left(t\right)={\displaystyle {\int }_D\left[\text{div}\left(C\left(T,T\right)\right)-\frac{n}{2}\text{div}\left({\Vert T\Vert }_g^{2}T\right)-\frac{1}{n}\Delta H-\frac{2\left(n+1\right)}{n}H\right]\phi \text{d}{V}_{M_t}}$

定理3. 对于变分向量场为 $f^{\prime }=\phi f$ 的正则变分，泛函 $ℭ={\displaystyle {\int }_D{\Vert C\Vert }^2\text{d}{V}_M}$ 的一阶变分为

${ℭ}^{\prime }\left(t\right)={\displaystyle {\int }_D\left[n^2\text{div}\left(C\left(T,T\right)\right)-\frac{n}{2}\text{div}\left({\Vert C\Vert }_g^{2}T\right)+\text{div}\left(C\left(\text{Ric}\right)\right)-n\Delta H-2n\left(n+1\right)H\right]\phi \text{d}{V}_{M_t}}$

容易验证满足 $\tilde{\nabla }T=0$ 的超曲面均是泛函 $\mathfrak{T}$ 和 $ℭ$ 的临界点，包括中心在原点的椭球面以及双曲型仿射球。上述泛函的Euler-Lagrange方程以及临界超曲面值得进一步研究。

2. 参数化超曲面的中心仿射微分几何及若干基本几何量的变分公式

设 $M=f\left(D\right)$ 为 $R^{n+1}$ 中嵌入超曲面，其中 $f:D\to R^{n+1}$ 是非退化嵌入映射， $D\subseteq R^n$ 是连通开域。设 $f\left(p\right)$ 为超曲面 $M:=f\left(D\right)$ 上的点，其中 $p\in D$，那么切空间 $T_{f\left(p\right)}M$ 的基底为 $\left\{{\frac{\partial f}{\partial x^1}|}_p,\cdots ,{\frac{\partial f}{\partial x^n}|}_p\right\}$ 张成。由中心法化的条件 [1] [2] [3] 可知向量组

$\left\{{\frac{\partial f}{\partial x^1}|}_p,\cdots ,{\frac{\partial f}{\partial x^n}|}_p,-f\left(p\right)\right\}$ (1)

在M上处处线性无关，由(1)可得超曲面的基本方程

$\frac{{\partial }^{2}f}{\partial x^i\partial x^j}={\Gamma }_{ij}^l\frac{\partial f}{\partial x^l}-g_{ij}f$ (2)

令 $\nabla \frac{\partial f}{\partial x^i}:={\Gamma }_{ij}^l\text{d}{x}^j\otimes \frac{\partial f}{\partial x^l}$，那么容易验证 $\nabla$ 为M上的无挠仿射联络；令 $g:=g_{ij}\text{d}{x}^i\otimes \text{d}{x}^j$，易知 $g$ 为M上的半黎曼度量。在本文中，我们只考虑 $g$ 是正定的情形，此时 $g$ 为黎曼度量，称为M的中心仿射度量。

设 $\tilde{\nabla }$ 为度量 $g$ 的Levi-Civita联络， ${\tilde{\Gamma }}_{ij}^l$ 为 $\tilde{\nabla }$ 在自然标架(1)下的系数。令 $C=\nabla -\tilde{\nabla }$，则张量 $C$ 在自然标架(1)下表示为

$C_{ij}^k={\Gamma }_{ij}^k-{\tilde{\Gamma }}_{ij}^k$ (3)

本文中关于张量指标的上升和下降均由中心仿射度量 $g$ 诱导，例如， $C_{ijk}:=g_{il}{C}_{jk}^l$ 等。张量关于联络 $\tilde{\nabla }$ 的协变导数用“，”表示，关于联络 $\nabla$ 的协变导数用“；”表示，例如， ${\psi }_{;k}^l:=\frac{\partial {\psi }^l}{\partial x^k}+{\psi }^s{\Gamma }_{sk}^l$， $C_{ijk,l}:=\frac{\partial C_{ijk}}{\partial x^l}-C_{sjk}{\tilde{\Gamma }}_{il}^s-C_{isk}{\tilde{\Gamma }}_{jl}^s-C_{ijs}{\tilde{\Gamma }}_{kl}^s$，等。

设 $f:D\times \left(-\epsilon ,\epsilon \right)\to R^{n+1}$ 是超曲面 $f:D\to R^{n+1}$ 的适当变分，其变分向量场在自然标架场(1)下，可表示为

$f^{\prime }=\phi f+{\psi }^i\frac{\partial f}{\partial x^i}=:\phi f+\psi$ (4)

其中 $\phi$ 和 ${\psi }^i$ 是D上的光滑函数， $i=1,\cdots ,n$，并且在D的边界为零。

引理1. 中心仿射度量的变分公式以及仿射联络的变分公式如下

$\frac{\partial g_{ij}}{\partial t}=-{\phi }_{,ij}+{\phi }_{,l}{C}_{ij}^l+{\psi }_{j,i}+{\psi }_{i,j}$ (5)

$\frac{\partial {\Gamma }_{ij}^l}{\partial t}={\phi }_{,i}{\delta }_j^l+{\phi }_{,j}{\delta }_i^l+{\psi }^l{}_{;ij}+{\psi }^k\left(g_{ij}{\delta }_k^l-g_{ik}{\delta }_j^l\right)$ (6)

证明：对(2)式关于t求导数，整理可得

$\begin{array}{c}\frac{\partial }{\partial t}\frac{{\partial }^{2}f}{\partial x^i\partial x^j}=\frac{\partial }{\partial t}\left({\Gamma }_{ij}^l\frac{\partial f}{\partial x^l}-g_{ij}f\right)\\ =\frac{\partial {\Gamma }_{ij}^l}{\partial t}\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^l\frac{{\partial }^{2}f}{\partial t\partial x^l}-\frac{\partial g_{ij}}{\partial t}f-\frac{\partial f}{\partial t}\\ =\frac{\partial {\Gamma }_{ij}^l}{\partial t}\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^l\frac{\partial }{\partial x^l}\left(\phi f+{\psi }^k\frac{\partial f}{\partial x^k}\right)-\frac{\partial g_{ij}}{\partial t}f-g_{ij}\left(\phi f+{\psi }^k\frac{\partial f}{\partial x^k}\right)\\ =\frac{\partial {\Gamma }_{ij}^l}{\partial t}\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^l{\phi }_{,l}f+\phi {\Gamma }_{ij}^l\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^l\frac{\partial {\psi }_{}^k}{\partial x^l}\frac{\partial f}{\partial x^k}+{\psi }^k{\Gamma }_{ij}^l\frac{{\partial }^{2}f}{\partial x^l\partial x^k}-\frac{\partial g_{ij}}{\partial t}f-g_{ij}\left(\phi f+{\psi }^k\frac{\partial f}{\partial x^k}\right)\\ =\left(\frac{\partial {\Gamma }_{ij}^l}{\partial t}+\phi {\Gamma }_{ij}^l+\frac{\partial {\psi }_{}^l}{\partial x^k}{\Gamma }_{ij}^k+{\psi }^k{\Gamma }_{ij}^s{\Gamma }_{ks}^l-g_{ij}{\psi }^l\right)\frac{\partial f}{\partial x^l}+\left(-\frac{\partial g_{ij}}{\partial t}-\phi g_{ij}-{\psi }^k{\Gamma }_{ij}^l{g}_{kl}+{\phi }_{,l}{\Gamma }_{ij}^l\right)f\end{array}$ (7)

另一方面，对(4)关于 $x^i$， $x^j$ 求导，整理可得

$\begin{array}{c}\frac{{\partial }^2}{\partial x^i\partial x^j}\frac{\partial f}{\partial t}=\frac{{\partial }^2}{\partial x^i\partial x^j}\left(\phi f+{\psi }^k\frac{\partial f}{\partial x^k}\right)\\ =\frac{\partial }{\partial x^j}\left({\phi }_{,i}f+\phi \frac{\partial f}{\partial x^i}+\frac{\partial {\psi }^k}{\partial x^i}\frac{\partial f}{\partial x^k}+{\psi }^k\frac{{\partial }^{2}f}{\partial x^k\partial x^i}\right)\\ =\frac{{\partial }^2\phi }{\partial x^i\partial x^j}f+{\phi }_{,i}\frac{\partial f}{\partial x^j}+{\phi }_{,j}\frac{\partial f}{\partial x^i}+\phi \frac{{\partial }^{2}f}{\partial x^i\partial x^j}+\frac{{\partial }^2{\psi }^l}{\partial x^i\partial x^j}\frac{\partial f}{\partial x^l}+\frac{\partial {\psi }^k}{\partial x^i}\frac{{\partial }^{2}f}{\partial x^k\partial x^j}\\ +\frac{\partial {\psi }^k}{\partial x^j}\frac{{\partial }^{2}f}{\partial x^k\partial x^i}+{\psi }^k\frac{{\partial }^{3}f}{\partial x^k\partial x^i\partial x^j}\\ =\frac{{\partial }^2\phi }{\partial x^i\partial x^j}f+{\phi }_{,i}\frac{\partial f}{\partial x^j}+{\phi }_{,j}\frac{\partial f}{\partial x^i}+\left(\phi {\Gamma }_{ij}^l\frac{\partial f}{\partial x^l}-\phi g_{ij}f\right)+{\psi }^k\frac{{\partial }^{3}f}{\partial x^k\partial x^i\partial x^j}+\frac{{\partial }^2{\psi }^l}{\partial x^i\partial x^j}\frac{\partial f}{\partial x^l}\\ +\left(\frac{\partial {\psi }^k}{\partial x^i}{\Gamma }_{kj}^l\frac{\partial f}{\partial x^l}-\frac{\partial {\psi }^k}{\partial x^i}{g}_{kj}f\right)+\left(\frac{\partial {\psi }^k}{\partial x^j}{\Gamma }_{ki}^l\frac{\partial f}{\partial x^l}-\frac{\partial {\psi }^k}{\partial x^j}{g}_{ki}f\right)\end{array}$ (8)

对于(2)关于 $x^k$ 求导数得到

$\begin{array}{c}\frac{{\partial }^{3}f}{\partial x^k\partial x^i\partial x^j}=\frac{\partial }{\partial x^k}\left({\Gamma }_{ij}^l\frac{\partial f}{\partial x^l}-g_{ij}f\right)\\ =\frac{\partial {\Gamma }_{ij}^l}{\partial x^k}\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^l\frac{{\partial }^{2}f}{\partial x^l\partial x^k}-\frac{\partial g_{ij}}{\partial x^k}f-g_{ij}\frac{\partial f}{\partial x^k}\\ =\frac{\partial {\Gamma }_{ij}^l}{\partial x^k}\frac{\partial f}{\partial x^l}+{\Gamma }_{ij}^s{\Gamma }_{sk}^l\frac{\partial f}{\partial x^l}-{\Gamma }_{ij}^l{g}_{lk}f-\frac{\partial g_{ij}}{\partial x^k}f-g_{ij}\frac{\partial f}{\partial x^k}\\ =\left(\frac{\partial {\Gamma }_{ij}^l}{\partial x^k}+{\Gamma }_{ij}^s{\Gamma }_{sk}^l-g_{ij}{\delta }_k^l\right)\frac{\partial f}{\partial x^l}-\left(\frac{\partial g_{ij}}{\partial x^k}+{\Gamma }_{ij}^l{g}_{lk}\right)f\end{array}$ (9)

根据(9)式，以及 $\frac{{\partial }^{3}f}{\partial x^k\partial x^i\partial x^j}=\frac{{\partial }^{3}f}{\partial x^j\partial x^i\partial x^k}$，可得如下的可积性条件

$\frac{\partial {\Gamma }_{ij}^l}{\partial x^k}+{\Gamma }_{ij}^s{\Gamma }_{sk}^l-g_{ij}{\delta }_k^l=\frac{\partial {\Gamma }_{ik}^l}{\partial x^j}+{\Gamma }_{ik}^s{\Gamma }_{sj}^l-g_{ik}{\delta }_j^l$ (10)

将(9)式带入(8)式，并利用(10)式，得到

$\begin{array}{c}\frac{{\partial }^2}{\partial x^i\partial x^j}\frac{\partial f}{\partial t}=\left(\frac{{\partial }^2\phi }{\partial x^i\partial x^j}-\frac{\partial {\psi }^k}{\partial x^i}{g}_{kj}-\frac{\partial {\psi }^k}{\partial x^j}{g}_{ki}-{\psi }^k\left(\frac{\partial g_{ij}}{\partial x^k}+{\Gamma }_{ij}^l{g}_{lk}\right)-\phi g_{ij}\right)f\\ +[{\phi }_{,i}{\delta }_j^l+{\phi }_{,j}{\delta }_i^l+\phi {\Gamma }_{ij}^l+\frac{{\partial }^2{\psi }^l}{\partial x^i\partial x^j}+\frac{\partial {\psi }^k}{\partial x^i}{\Gamma }_{kj}^l+\frac{\partial {\psi }^k}{\partial x^j}{\Gamma }_{ki}^l\\ +{\psi }^k\left(\frac{\partial {\Gamma }_{ik}^l}{\partial x^j}+{\Gamma }_{ik}^s{\Gamma }_{sj}^l-g_{ik}{\delta }_j^l\right)]\frac{\partial f}{\partial x^l}\end{array}$ (11)

比较(7)和(11)两式，可得

$-\frac{\partial g_{ij}}{\partial t}-{\psi }^k{\Gamma }_{ij}^l{g}_{kl}+{\phi }_{,l}{\Gamma }_{ij}^l=\frac{{\partial }^2\phi }{\partial x^i\partial x^j}-\frac{\partial {\psi }^k}{\partial x^i}{g}_{kj}-\frac{\partial {\psi }^k}{\partial x^j}{g}_{ki}-{\psi }^k\left(\frac{\partial g_{ij}}{\partial x^k}+{\Gamma }_{ij}^l{g}_{lk}\right)$ (12)

以及

$\begin{array}{l}\frac{\partial {\Gamma }_{ij}^l}{\partial t}+\frac{\partial {\psi }^l}{\partial x^k}{\Gamma }_{ij}^k+{\psi }^k{\Gamma }_{ij}^s{\Gamma }_{ks}^l-g_{ij}{\psi }^l\\ ={\phi }_{,i}{\delta }_j^l+{\phi }_{,j}{\delta }_i^l+\frac{{\partial }^2{\psi }^l}{\partial x^i\partial x^j}+\frac{\partial {\psi }^k}{\partial x^i}{\Gamma }_{kj}^l+\frac{\partial {\psi }^k}{\partial x^j}{\Gamma }_{ki}^l+{\psi }^k\left(\frac{\partial {\Gamma }_{ik}^l}{\partial x^j}+{\Gamma }_{ik}^s{\Gamma }_{sj}^l-g_{ik}{\delta }_j^l\right)\end{array}$ (13)

进一步简化(12)式如下

$\begin{array}{c}-\frac{\partial g_{ij}}{\partial t}=\frac{{\partial }^2\phi }{\partial x^i\partial x^j}-{\phi }_{,l}{\Gamma }_{ij}^l-\frac{\partial {\psi }^k}{\partial x^i}-\frac{\partial {\psi }^k}{\partial x^j}{g}_{ki}-{\psi }^k\left(\frac{\partial g_{ij}}{\partial x^k}+{\Gamma }_{ij}^l{g}_{lk}\right)+{\psi }^k{\Gamma }_{ij}^l{g}_{kl}\\ =\frac{{\partial }^2\phi }{\partial x^i\partial x^j}-{\phi }_{,l}{\tilde{\Gamma }}_{ij}^l-{\phi }_{,l}{C}_{ij}^l-\frac{\partial {\psi }_j}{\partial x^i}-\frac{\partial {\psi }_i}{\partial x^j}+{\psi }^k\frac{\partial g_{kj}}{\partial x^i}+{\psi }^k\frac{\partial g_{ki}}{\partial x^j}-{\psi }_s{g}^{st}\frac{\partial g_{ij}}{\partial x^t}\\ ={\phi }_{,ij}-{\phi }_{,l}{C}_{ij}^l-\frac{\partial {\psi }_j}{\partial x^i}-\frac{\partial {\psi }_i}{\partial x^j}+{\psi }_s{g}^{st}\frac{\partial g_{tj}}{\partial x^i}+{\psi }_s{g}^{st}\frac{\partial g_{ti}}{\partial x^j}-{\psi }_s{g}^{st}\frac{\partial g_{ij}}{\partial x^t}\\ ={\phi }_{,ij}-{\phi }_{,l}{C}_{ij}^l-\frac{\partial {\psi }_j}{\partial x^i}-\frac{\partial {\psi }_i}{\partial x^j}+2{\psi }_s{\tilde{\Gamma }}_{ij}^s\\ ={\phi }_{,ij}-{\phi }_{,l}{C}_{ij}^l-{\psi }_{j,i}-{\psi }_{i,j}\end{array}$ (14)

直接计算 $\psi$ 关于仿射联络 $\nabla$ 的二阶协变导数，可知

$\begin{array}{c}{\psi }^l{}_{;ij}=\frac{\partial }{\partial x^j}\left(\frac{\partial {\psi }^l}{\partial x^i}+{\psi }^k{\Gamma }_{ki}^l\right)+\left(\frac{\partial {\psi }^s}{\partial x^i}+{\psi }^k{\Gamma }_{ki}^s\right){\Gamma }_{sj}^l-\left(\frac{\partial {\psi }^l}{\partial x^s}+{\psi }^k{\Gamma }_{ks}^l\right){\Gamma }_{ij}^s\\ =\frac{{\partial }^2{\psi }^l}{\partial x^i\partial x^j}+\frac{\partial {\psi }^k}{\partial x^j}{\Gamma }_{ki}^l+{\psi }^k\frac{\partial {\Gamma }_{ik}^l}{\partial x^j}+\frac{\partial {\psi }^s}{\partial x^i}{\Gamma }_{sj}^l+{\psi }^k{\Gamma }_{ik}^s{\Gamma }_{sj}^l-\frac{\partial {\psi }^l}{\partial x^s}{\Gamma }_{ij}^s-{\psi }^k{\Gamma }_{ks}^l{\Gamma }_{ij}^s\end{array}$ (15)

将(15)式代入(13)式，可得

$\frac{\partial {\Gamma }_{ij}^l}{\partial t}={\phi }_{,i}{\delta }_j^l+{\phi }_{,j}{\delta }_i^l+{\psi }^l{}_{;ij}+{\psi }^k\left(g_{ij}{\delta }_k^l-g_{ik}{\delta }_j^l\right)$ (16)

证毕。

引理2. 假设变分向量场的切向分量 $\psi$ 为零，那么中心仿射度量g的Levi-Civita联络 $\tilde{\nabla }$ 以及Techebychev形式T变分公式如下

$\frac{\partial {\tilde{\Gamma }}_{jk}^i}{\partial t}=-\frac{1}{2}{g}^{il}\left[{\left({\phi }_{,kl}-{\phi }_{,s}{C}_{kl}^s\right)}_{,j}+{\left({\phi }_{,jl}-{\phi }_{,s}{C}_{jl}^s\right)}_{,k}-{\left({\phi }_{,jk}-{\phi }_{,s}{C}_{jk}^s\right)}_{,l}\right]$ (17)

$n\frac{\partial T_j}{\partial t}=\left(n+1\right){\phi }_{,j}+\frac{1}{2}{\left(\Delta \phi \right)}_{,j}-\frac{1}{2}{\left(n{\phi }_{,s}{T}^s\right)}_{,j}$ (18)

证明：下面Levi-Civita联络的变分公式 [13] 是容易证明的

$\frac{\partial {\tilde{\Gamma }}_{jk}^i}{\partial t}=\frac{1}{2}{g}^{il}\left[{\left(\frac{\partial g_{kl}}{\partial t}\right)}_{,j}+{\left(\frac{\partial g_{jl}}{\partial t}\right)}_{,k}-{\left(\frac{\partial g_{jk}}{\partial t}\right)}_{,l}\right]$ (19)

将(5)代入(19)，立即得到(17)。取 $\frac{\partial \tilde{\nabla }}{\partial t}$ 的迹得到

$\frac{\partial {\tilde{\Gamma }}_{sk}^k}{\partial t}=\frac{1}{2}{g}^{kl}{\left(\frac{\partial g_{kl}}{\partial t}\right)}_{,s}=\frac{1}{2}{g}^{kl}{\left(-{\phi }_{kl}+{\phi }_{,q}{C}_{kl}^q\right)}_{,s}=\frac{1}{2}{\left(-\Delta \phi +n{\phi }_{,q}{T}^q\right)}_{,s}$ (20)

根据条件 $\psi =0$ 的假设，以及(3)和(20)式，可知

$n\frac{\partial T_j}{\partial t}=\frac{\partial C_{ji}^i}{\partial t}=\frac{\partial {\Gamma }_{ji}^i}{\partial t}-\frac{\partial {\tilde{\Gamma }}_{ji}^i}{\partial t}=\left(n+1\right){\phi }_{,j}+\frac{1}{2}{\left(\Delta \phi \right)}_{,j}-\frac{1}{2}{\left(n{\phi }_{,s}{T}^s\right)}_{,j}$ (21)

证毕。

3. 体积的第一变分公式与第二变分公式

在本节，我们计算超曲面M的中心仿射体积的变分公式，从而给出定理1的证明。

定理1的证明：

根据M的中心仿射体积定义 $V={\displaystyle {\int }_D\text{d}{V}_M}={\displaystyle {\int }_D\sqrt{\det \left(g_{ij}\right)}}\text{d}{x}^1\wedge \cdots \wedge \text{d}{x}^n$，可知

$V^{\prime }\left(t\right)={\displaystyle {\int }_D\frac{\partial }{\partial t}\text{d}{V}_{M_t}}=\frac{1}{2}{\displaystyle {\int }_D\frac{1}{\sqrt{\det \left(g_{ij}\right)}}}\frac{\partial \det \left(g_{ij}\right)}{\partial t}\text{d}{x}^1\wedge \cdots \wedge \text{d}{x}^n=\frac{1}{2}{\displaystyle {\int }_D{g}^{ij}\frac{\partial g_{ij}}{\partial t}\text{d}{V}_{M_t}}$ (22)

将(5)式代入(22)式，并利用散度定理，可得

$\begin{array}{c}{V}^{\prime }\left(t\right)=-\frac{1}{2}{\displaystyle {\int }_D{g}^{ij}\left({\phi }_{,ij}-{\phi }_{,l}{C}_{ij}^l-{\psi }_{j,i}-{\psi }_{i,j}\right)\text{d}{V}_{M_t}}\\ =-\frac{1}{2}{\displaystyle {\int }_D\left(\Delta \phi -2\text{div}\psi -n{\phi }_{,l}{T}^l\right)\text{d}{V}_{M_t}}\\ =-\frac{n}{2}{\displaystyle {\int }_D\phi T_{,k}^k}\text{d}{V}_{M_t}\end{array}$ (23)

根据平均曲率 $H=d^{*}T=T_{,k}^k$ 的定义，可知(1)式成立。

接下来计算中心仿射极小超曲面体积的第二变分公式。由于变分向量场中的切向分量 $\psi$ 对第一变分公式没有贡献，为简单起见，我们假设 $\psi =0$，根据第一变分公式(23)以及 $H=0$ 的假设，我们只需计算 $\frac{\partial H}{\partial t}=\frac{\partial T_{,k}^k}{\partial t}$，其中 $T_{,k}^l=\frac{\partial T^l}{\partial x^k}+T^s{\tilde{\Gamma }}_{sk}^l$。

根据(5)和(18)式，可知

$\begin{array}{c}\frac{\partial T^k}{\partial t}=\frac{\partial g^{kj}}{\partial t}{T}_j+g^{kj}\frac{\partial T_j}{\partial t}=-g^{kh}\frac{\partial g_{hz}}{\partial t}{g}^{zj}{T}_j+g^{kj}\frac{\partial T_j}{\partial t}\\ =g^{kh}\left({\phi }_{,hz}{T}^z-{\phi }_{,q}{C}_{hz}^q{T}^z\right)+\frac{1}{2n}{g}^{kj}{\left[\Delta \phi -n{\phi }_{,q}{T}^q+2\left(n+1\right)\phi \right]}_{,j}\end{array}$ (24)

由(24)式，我们可以得到

${\left(\frac{\partial T^k}{\partial t}\right)}_{,l}=g^{kh}{\left({\phi }_{,hz}{T}^z-{\phi }_{,q}{C}_{hz}^q{T}^z\right)}_{,l}+\frac{1}{2n}{g}^{kj}{\left[\Delta \phi -n{\phi }_{,q}{T}^q+2\left(n+1\right)\phi \right]}_{,jl}$ (25)

根据熟知的公式 $\frac{\partial }{\partial t}\left(\tilde{\nabla }T\right)=\tilde{\nabla }\frac{\partial T}{\partial t}+\frac{\partial \tilde{\nabla }}{\partial t}T$，以及对(25)式取迹，可知

$\begin{array}{c}\frac{\partial T_{,k}^k}{\partial t}={\left(\frac{\partial T^k}{\partial t}\right)}_{,k}+T^k\frac{\partial {\tilde{\Gamma }}_{ks}^s}{\partial t}\\ =g^{kh}{\left({\phi }_{,hz}{T}^z-{\phi }_{,q}{C}_{hz}^q{T}^z\right)}_{,k}+\frac{1}{2n}\Delta \left[\Delta \phi -n{\phi }_{,q}{T}^q+2\left(n+1\right)\phi \right]\\ +\frac{T^k}{2}{\left(-\Delta \phi +n{\phi }_{,q}{T}^q\right)}_{,k}\end{array}$ (26)

根据(23)和(26)式，运用散度定理，我们得到中心仿射极小超曲面的体积第二变分公式

$\begin{array}{c}V{\left(0\right)}^{\prime \text{}\prime }=-\frac{n}{2}{\displaystyle {\int }_D\frac{\partial T_{,k}^k}{\partial t}\phi \text{d}{V}_M}=-\frac{n}{2}{\displaystyle {\int }_D\left[{\left(\frac{\partial T^k}{\partial t}\right)}_{,k}+T^k\frac{\partial {\tilde{\Gamma }}_{ks}^s}{\partial t}\right]\phi \text{d}{V}_M}\\ =-\frac{n}{2}{\displaystyle {\int }_D\left[g^{kh}{\left({\phi }_{,hz}{T}^z-{\phi }_{,q}{C}_{hz}^q{T}^z\right)}_{,k}+\frac{1}{2n}\Delta \left[\Delta \phi -n{\phi }_{,q}{T}^q+2\left(n+1\right)\phi \right]+\frac{1}{2}{T}^k{\left(-\Delta \phi +n{\phi }_{,q}{T}^q\right)}_{,k}\right]\phi \text{d}{V}_M}\\ =-\frac{n}{2}{\displaystyle {\int }_D\left[-g^{kh}\left({\phi }_{,hz}{T}^z-{\phi }_{,q}{C}_{hz}^q{T}^z\right){\phi }_{,k}+\frac{1}{2n}\Delta \phi \left[\Delta \phi -n{\phi }_{,q}{T}^q+2\left(n+1\right)\phi \right]-\frac{1}{2}{\phi }_{,k}{T}^k\left(-\Delta \phi +n{\phi }_{,q}{T}^q\right)\right]\text{d}{V}_M}\\ =-\frac{1}{4}{\displaystyle {\int }_D\left[-2n{g}^{kh}{\phi }_{,hz}{T}^z{\phi }_{,k}+2n{g}^{kh}{\phi }_{,q}{C}_{hz}^q{T}^z{\phi }_{,k}+\Delta \phi \left[\Delta \phi +2\left(n+1\right)\phi \right]-{\left(n{\phi }_{,q}{T}^q\right)}^2\right]\text{d}{V}_M}\\ =-\frac{1}{4}{\displaystyle {\int }_D\left[-n\text{div}\left({\Vert \nabla \phi \Vert }_g^{2}T\right)+\Delta \phi \left[\Delta \phi +2\left(n+1\right)\phi \right]-{\left(nT\left(\nabla \phi \right)\right)}^2+2nC\left(\nabla \phi ,\nabla \phi \right)\left(T\right)\right]\text{d}{V}_M}\\ =-\frac{1}{4}{\displaystyle {\int }_D\left[\Delta \phi \left[\Delta \phi +2\left(n+1\right)\phi \right]-{\left(nT\left(\nabla \phi \right)\right)}^2+2nC\left(\nabla \phi ,\nabla \phi \right)\left(T\right)\right]\text{d}{V}_M}\end{array}$

证毕。

4. Techebychev形式长度平方积分以及3形式长度平方积分的变分公式

本节我们首先计算Techebychev形式长度平方 ${\Vert T\Vert }_g^2$ 的变分公式，然后计算其积分的变分公式，从而给出定理2的证明。然后计算全曲率的变分公式，并利用仿射Gauss公式，给出3形式的长度平方积分的变分公式。在本节，我们假设变分向量场的切向分量 $\psi$ 为零。

引理3. Techebychev形式长度平方 ${\Vert T\Vert }_g^2$ 的变分公式如下

$\frac{\partial }{\partial t}{\Vert T\Vert }_g^2=-T^i{T}^j\left(-{\phi }_{,ij}+{\phi }_{,l}{C}_{ij}^l\right)+\frac{1}{n}{T}^j{\left[2\left(n+1\right)\phi +\Delta \phi -\left(n{\phi }_{,s}{T}^s\right)\right]}_{,j}$ (27)

证明：根据定义 ${\Vert T\Vert }_g^2=g^{ij}{T}_i{T}_j$，可得

$\frac{\partial }{\partial t}{\Vert T\Vert }_g^2=\frac{\partial g^{ij}}{\partial t}{T}_i{T}_j+2g^{ij}{T}_i\frac{\partial T_j}{\partial t}$ (28)

将(5)式和(18)式，代入(28)式，便可得到(27)式。

证毕。

定理2的证明：根据(5)、(22)和(27)式，可知

$\begin{array}{c}{\mathfrak{T}}^{\prime }\left(t\right)={\displaystyle {\int }_D\frac{\partial {\Vert T\Vert }_g^2}{\partial t}\text{d}{V}_{M_t}}+{\displaystyle {\int }_D{\Vert T\Vert }_g^2\frac{\partial }{\partial t}\text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D{T}^i{T}^j\left({\phi }_{,ij}-{\phi }_{,l}{C}_{ij}^l\right)+\frac{1}{n}{T}^j{\left[2\left(n+1\right)\phi +\Delta \phi -\left(n{\phi }_{,s}{T}^s\right)\right]}_{,j}\text{d}{V}_{M_t}}-\frac{1}{2}{\displaystyle {\int }_D{\Vert T\Vert }_g^2\left(\Delta \phi -n{\phi }_{,l}{T}^l\right)\text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D\left[{\left(T^i{T}^j\right)}_{,ij}+{\left(T^i{T}^j{C}_{ij}^l\right)}_{,l}-\frac{2\left(n+1\right)}{n}H-\frac{1}{n}\Delta H-{\left(H{T}^s\right)}_{,s}\right]\phi \text{d}{V}_{M_t}}-\frac{1}{2}{\displaystyle {\int }_D\left[\Delta {\Vert T\Vert }_g^2+n\text{div}\left({\Vert T\Vert }_g^{2}T\right)\right]\phi \text{d}{V}_{M_t}}\end{array}$

$\begin{array}{l}={\displaystyle {\int }_D\left[\left({\left(H{T}^j\right)}_{,j}+T_{,j}^i{T}_{,i}^j+T^i{T}_{,ij}^j\right)+{\left(T^i{T}^j{C}_{ij}^l\right)}_{,l}-\frac{2\left(n+1\right)}{n}H-\frac{1}{n}\Delta H-{\left(H{T}^s\right)}_{,s}\right]\phi \text{d}{V}_{M_t}}\\ -\frac{1}{2}{\displaystyle {\int }_D\left[\Delta {\Vert T\Vert }_g^2+n\text{div}\left({\Vert T\Vert }_g^{2}T\right)\right]\phi \text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D\left[{\Vert \tilde{\nabla }T\Vert }_g^2+T^i{T}_{i,jj}+{\left(T^i{T}^j{C}_{ij}^l\right)}_{,l}-\frac{2\left(n+1\right)}{n}H-\frac{1}{n}\Delta H-\frac{1}{2}\Delta {\Vert T\Vert }_g^2-\frac{n}{2}\text{div}\left({\Vert T\Vert }_g^{2}T\right)\right]\phi \text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D\left[{\Vert \tilde{\nabla }T\Vert }_g^2+T^i\Delta T_i+\text{div}\left(C\left(T,T\right)\right)-\frac{2\left(n+1\right)}{n}H-\frac{1}{n}\Delta H-\frac{1}{2}\Delta {\Vert T\Vert }_g^2-\frac{n}{2}\text{div}\left({\Vert T\Vert }_g^{2}T\right)\right]\phi \text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D\left[\text{div}\left(C\left(T,T\right)\right)-\frac{n}{2}\text{div}\left({\Vert T\Vert }_g^{2}T\right)-\frac{1}{n}\Delta H-\frac{2\left(n+1\right)}{n}H\right]\phi \text{d}{V}_{M_t}}\end{array}$

证毕。

引理4. $M$ 的全数量曲率积分泛函 $S={\displaystyle {\int }_D\text{scal}\text{d}{V}_M}$ 的变分公式如下

$S^{\prime }\left(t\right)={\displaystyle {\int }_D\left[\text{div}\left(C\left(\text{Ric}\right)\right)-\frac{n}{2}\text{div}\left(\text{scal}\cdot T\right)\right]\phi \text{d}{V}_{M_t}}$ (29)

其中Ric和scal分别表示 $M$ 的Ricci曲率张量和数量曲率。

证明：根据数量曲率的一般变分公式 [13] ，以及定理1的证明，可知

$\begin{array}{c}{S}^{\prime }\left(t\right)=\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_D\text{scal}\text{d}{V}_{M_t}}={\displaystyle {\int }_D\frac{\partial \text{scal}}{\partial t}\text{d}{V}_{M_t}}+{\displaystyle {\int }_D\text{scal}\frac{\partial }{\partial t}\text{d}{V}_{M_t}}\\ =-{\displaystyle {\int }_D{\text{Ric}}^{ij}\frac{\partial g_{ij}}{\partial t}\text{d}{V}_{M_t}}+\frac{1}{2}{\displaystyle {\int }_D\text{scal}\cdot {\text{g}}^{ij}\frac{\partial g_{ij}}{\partial t}\text{d}{V}_{M_t}}\end{array}$ (30)

将(5)式代入(30)式，经过分部积分可得

$\begin{array}{c}{S}^{\prime }\left(t\right)={\displaystyle {\int }_D\left(\frac{1}{2}\text{scal}\cdot {\text{g}}^{ij}-{\text{Ric}}^{ij}\right)\left(-{\phi }_{ij}+{\phi }_s{C}_{ij}^s\right)\text{d}{V}_{M_t}}\\ ={\displaystyle {\int }_D\left[-\frac{1}{2}\Delta \text{scal}-\frac{n}{2}{\left(\text{scal}\cdot T^s\right)}_{,s}+{\text{Ric}}_{,ji}^{ij}+{\left({\text{Ric}}^{ij}{C}_{ij}^s\right)}_{,s}\right]\phi \text{d}{V}_{M_t}}\end{array}$ (31)

根据公式 [14] $\text{d}\text{scal}=2\text{div}\left(\text{Ric}\right)$，可知 ${\text{Ric}}_{,ji}^{ij}=\frac{1}{2}\Delta \text{scal}$，代入(31)式，可得(29)式。

证毕。

定理3的证明：

根据仿射Gauss方程 [1] [2] [3] 可知

$\text{scal}=n\left(n-1\right)+{\Vert C\Vert }_g^2-n^2{\Vert T\Vert }_g^2$ (31)

因此

$ℭ=S-n\left(n-1\right)V+n^2\mathfrak{T}$ (32)

对(32)求导，并利用定理1，引理4以及定理2的结论以及(31)式，可知

$\begin{array}{c}{ℭ}^{\prime }\left(t\right)=S^{\prime }\left(t\right)-n\left(n-1\right)V^{\prime }\left(t\right)+n^2{\mathfrak{T}}^{\prime }\left(t\right)\\ ={\displaystyle {\int }_D\left[\text{div}\left(C\left(\text{Ric}\right)\right)-\frac{n}{2}\text{div}\left(\text{scal}\cdot T\right)\right]\phi \text{d}{V}_{M_t}}+\frac{n^2\left(n-1\right)}{2}{\displaystyle {\int }_{D}H\phi \text{d}{V}_{M_t}}\\ +n^2{\displaystyle {\int }_D\left[\text{div}\left(C\left(T,T\right)\right)-\frac{n}{2}\text{div}\left({\Vert T\Vert }_g^{2}T\right)-\frac{1}{n}\Delta H-\frac{2\left(n+1\right)}{n}H\right]\phi \text{d}{V}_{M_t}}\\ =n^2{\displaystyle {\int }_D\left[\text{div}\left(C\left(T,T\right)\right)-\frac{1}{2n}\text{div}\left({\Vert C\Vert }_g^{2}T\right)+\frac{1}{n^2}\text{div}\left(C\left(\text{Ric}\right)\right)-\frac{1}{n}\Delta H-\frac{2\left(n+1\right)}{n}H\right]\phi \text{d}{V}_{M_t}}\end{array}$

证毕。

致谢

非常感谢我的导师——李明副教授在本论文研究方面的指导和帮助。

基金项目

国家自然科学基金面上项目(编号：11871126)。