Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

Abstract

We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension $n+1\ge 3$. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schn\"urer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.