- Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space(arXiv)
Author : Yimin Chen
Abstract : In this paper, we prove the Heintze-Karcher’s type inequality for capillary hypersurfaces supported on a totally geodesic hyper-plane in hyperbolic space, and equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this result to prove the Alexandrov type theorem for embedded capillary hypersurfaces in Hn+1+. In addiction, we prove some other rigidity results for immersed capillary hypersurfaces.
2. Reverse Faber-Krahn inequality for the p-Laplacian in Hyperbolic space(arXiv)
Abstract : In this paper, we study the shape optimization problem for the first eigenvalue of the p-Laplace operator with the mixed Neumann-Dirichlet boundary conditions on multiply-connected domains in hyperbolic space. Precisely, we establish that among all multiply-connected domains of a given volume and prescribed (n−1)-th quermassintegral of the convex Dirichlet boundary (inner boundary), the concentric annular region produces the largest first eigenvalue. We also derive Nagy’s type inequality for outer parallel sets of a convex domain in the hyperbolic space
3.Discrete-time gradient flows in Gromov hyperbolic spaces (arXiv)
Author : Shin-ichi Ohta
Abstract : We investigate fundamental properties of the proximal point algorithm for Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces. We show that the proximal point algorithm from an arbitrary initial point can find a point close to a minimizer of the function. Moreover, we establish a contraction estimate for the proximal (resolvent) operator.