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

TL;DR

This paper studies the stability of translating solitons in hyperbolic space under mean curvature flow, showing horospheres are stable and constructing new rotationally invariant solutions as barriers.

Contribution

It introduces a comprehensive theory for translating solitons in hyperbolic space, including stability results and new barrier solutions analogous to winglike solitons.

Findings

Horospheres are dynamically stable as solutions to MCF in hyperbolic space.

Constructed rotationally invariant translators serving as barriers.

Applied maximum principles and avoidance principles in the analysis.

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.