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.

中文翻译：

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一个积分公式及其在亚静态流形上的应用

在本文中，我们首先建立主要工具——具有多个边界分量（或无边界）的黎曼流形的积分公式。这个公式概括了来自 \cite{Re2} 的 Reilly 原始公式和来自 \cite{QX} 的最新结果。无论底层拓扑如何，它都为亚静态流形提供了强大的工具。使用这个公式和合适的椭圆偏微分方程，我们证明了一般亚静态流形中的有界域的 Heintze-Karcher 类型不等式，它从 Brendle \cite{Br} 中恢复了一些作为特殊情况的结果。另一方面，我们证明了亚静态翘曲积流形中静态凸超曲面的 Minkowski 不等式。此外，对于双曲空间中的horo-凸面超曲面和半球中的凸超曲面，我们获得了一个几乎 Schur 引理，