Rigidity of Solitons to the Mean Curvature Flow in $\mathbb{H}^3$ as Translation Surfaces

TL;DR

This paper investigates the rigidity properties of translation surfaces in hyperbolic 3-space, focusing on minimal surfaces and solitons to the mean curvature flow, revealing conditions under which these surfaces are uniquely determined.

Contribution

It introduces new rigidity results for translation surfaces in hyperbolic space related to minimal surfaces and mean curvature flow solitons, expanding understanding of their geometric properties.

Findings

Rigidity results for certain translation surfaces in hyperbolic space

Characterization of solitons to the mean curvature flow in this setting

Conditions under which these surfaces are uniquely determined

Abstract

In the half-space model of the hyperbolic three space with the hyperbolic metric, this same space can be seen as the Lie group, hence, a translation surface is a surface that is given by the product of two curves $\alpha$ and $\beta$ in this group. Here we present the rigidity of this kind of surfaces for some particular products in the context of minimal surfaces and solitons to the Mean Curvature Flow, also known as self-similar solutions.