In this article, we first establish the main tool—an integral formula (1.1) for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly’s original formula from  and the recent result from . It provides a robust tool for sub-static manifolds regardless of the underlying topology.
Using (1.1) 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  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.
Research of CX is supported in part by NSFC (Grant No. 11501480) and the Natural Science Foundation of Fujian Province of China (Grant No. 2017J06003).
"An integral formula and its applications on sub-static manifolds." J. Differential Geom. 113 (3) 493 - 518, November 2019. https://doi.org/10.4310/jdg/1573786972