A partially overdetermined problem in domains with partial umbilical boundary in space forms

Abstract

In this paper, we consider a partially overdetermined mixed boundary value problem in space forms. We generalize the main result in [10] into the case of general domains with partial umbilical boundary in space forms. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is part of a geodesic ball.

Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space

“…For free boundary hypersurface supported on totally geodesic hyperplane in space form, Pyo proved a Heintze-Karcher type inequality in [Pyo19], and for free boundary case supporting on horospheres, Guo and Xia proved the inequality in [GX20].…”

“…Since there are Heintze-Karcher type inequalities for capillary hypersurfaces, Wang and Xia [WX19] prove the Alexandrov type theorem for free boundary hypersurface in geodesic in space forms. Also, there are some results for free boundary hypersurface such as horospheres and equidistant hypersurfaces in hyperbolic space by Guo and Xia [GX20] and totally geodesic hyperplane in space form by Pyo [Pyo19]. Jia, Xia and Zhang [JXZ22] prove a Alexandrov type theorem for capillary hypersurface in a half Euclidean space and a Euclidean ball.…”