Heintze-Karcher’s inequality is an optimal geometric inequality for embedded closed hypersurfaces, which can be used to prove Alexandrov’s soap bubble theorem on embedded closed CMC hypersurfaces in the Euclidean space. In this talk, we introduce a Heintze-Karcher-type inequality for hypersurfaces with boundary in convex domains. As application, we give a new proof of Wente’s Alexandrov-type theorem for embedded CMC capillary hypersurfaces in the half-space. Moreover, the proof can be adapted to the anisotropic case in the convex cone, which enable us to prove Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces in the convex cone.

This is based on joint works with Xiaohan Jia, Guofang Wang and Xuwen Zhang.