次静态流形上的一个积分公式及其应用
An integral formula and its applications on sub-static manifolds
摘要：
在这篇文章中，我们首先建立了主要工具--具有多重边界分支（或无边界）的黎曼流形的积分公式。这个公式推广了Reilly从\cite{Re2}得到的原始公式和从\cite{QX}得到的最新结果。它提供了一个强大的工具，子静态流形，无论底层的拓扑结构。利用这个公式和适当的椭圆型偏微分方程，我们证明了一般次静态流形中有界域的Heintze-Karcher型不等式，作为特例恢复了Brendle \cite{Br}的一些结果.另一方面，证明了亚静态曲积流形中静态凸超曲面的Minkowski不等式。此外，我们还得到了双曲空间中的次凸超曲面和半球面中的凸超曲面的一个几乎Schur引理，它可以看作是一个特殊的Alexandrov-Fenchel不等式.
英文摘要：
In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless of the underlying topology. Using this formula and suitable elliptic PDEs, we prove Heintze-Karcher type inequalities for bounded domains in general sub-static manifolds which recovers some of the results from Brendle \cite{Br} as special cases. On the other hand, we prove a Minkowski inequality for static convex hypersurfaces in a sub-static warped product manifold. Moreover, we obtain an almost Schur lemma for horo-convex hypersurfaces in the hyperbolic space and convex hypersurfaces in the hemi-sphere, which can be viewed as a special Alexandrov-Fenchel inequality.
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