In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. Our result also provides a proof of a conjecture proposed by Sternberg-Zumbrun in J Reine Angew Math 503 (1998), 63-85. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.Part of this work was done while CX was visiting the mathematical institute of Albert-Ludwigs-Universität Freiburg. He would like to thank the institute for its hospitality.

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space H n , by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.2010 Mathematics Subject Classification. 52A40, 53C65, 53C44. Key words and phrases. quermassintegral, curvature integral, isoperimetric problem, Alexandrov-Fenchel inequality.GW is partly supported by SFB/TR71 "Geometric partial differential equations" of DFG. Part of this work was done while CX was visiting the mathematical institute of Albert-Ludwigs-Universität Freiburg. He would like to thank the institute for its hospitality.

In this paper, we prove a generalization of Reilly's formula in [10]. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below.

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