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

Abstract

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings.

Dihedral rigidity in hyperbolic 3-space

“…Note that a 1 ∈ [−1, 1] and the case of a 1 = ±1 is used in the works of Jang-Miao [JM21] and the author [Cha21] to evaluate the hyperbolic mass (note that (B.3) was actually first observed by Guo-Xia [GX20], and it predates [JM21] and [Cha21]). Each face F is umbilic and the mean curvature is then H = −2a 1 by Lemma B.1.…”