Volume preserving non homogeneous mean curvature flow in hyperbolic space

2016/12/15 by Maria Chiara Bertini, Bertini, Maria Chiara, Giuseppe Pipoli +1
Mathematics · #35B40 #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1612.05288

openalex publication_date 2016/12/15 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.

Related

• Volume preserving mean curvature flow in the Hyperbolic space
• Volume preserving non homogeneous mean curvature flow of convex hypersurfaces
• Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space
• Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space
• Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space