Conformal solitons for the mean curvature flow in hyperbolic space

TL;DR

This paper classifies and analyzes conformal solitons, specifically horo-expanders, for the mean curvature flow in hyperbolic space, establishing existence, uniqueness, and rigidity results for various symmetric solutions.

Contribution

It provides a classification of symmetric conformal solitons in hyperbolic space and solves boundary value problems with sharp conditions for existence.

Findings

Classification of cylindrical and rotationally symmetric solitons.

Existence results for Plateau and Dirichlet problems at infinity.

Rigidity theorems for bowl and grim-reaper cylinders.

Abstract

In this paper we study conformal solitons for the mean curvature flow in hyperbolic space $\mathbb{H}^{n+1}$. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field $-\partial_0$. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of $\partial_{\infty}\mathbb{H}^{n+1}$. We conclude by proving rigidity results for bowl and grim-reaper cylinders.