An integral formula and its applications on sub-static manifolds
An integral formula and its applications on sub-static manifolds
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DOI：
10.4310/jdg/1573786972
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发表时间：
2016-03
影响因子：
2.5
通讯作者：
Junfang Li;C. Xia
中科院分区：
文献类型：
--
作者：
Junfang Li;C. Xia
在这篇文章中，我们首先建立了黎曼流形的主要工具—具有多个边界分量（或没有边界）的积分公式。这个公式推广了Reilly的原始公式\cite{Re2}和最近的结果\cite{QX}。无论底层拓扑如何，它都为子静态流形提供了一个健壮的工具。利用该公式和合适的椭圆偏微分方程，证明了一般亚静流形有界区域的Heintze-Karcher型不等式，并恢复了Brendle \cite{Br}的一些结果作为特例。另一方面，我们证明了亚静翘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凸-凸超曲面和半球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。
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.