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November 2019 An integral formula and its applications on sub-static manifolds
Junfang Li, Chao Xia
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J. Differential Geom. 113(3): 493-518 (November 2019). DOI: 10.4310/jdg/1573786972


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 [15] and the recent result from [17]. 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 [2] 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.

Funding Statement

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).


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Junfang Li. Chao Xia. "An integral formula and its applications on sub-static manifolds." J. Differential Geom. 113 (3) 493 - 518, November 2019.


Received: 3 September 2019; Published: November 2019
First available in Project Euclid: 15 November 2019

zbMATH: 07130529
MathSciNet: MR4031740
Digital Object Identifier: 10.4310/jdg/1573786972

Rights: Copyright © 2019 Lehigh University


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Vol.113 • No. 3 • November 2019
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