Graphical translators for anisotropic and crystalline mean curvature flow

TL;DR

This paper studies graphical translators in anisotropic mean curvature flow, providing explicit solutions and analyzing their properties under symmetry assumptions, advancing understanding of solitons in geometric flows.

Contribution

It explicitly constructs and characterizes graphical translators in anisotropic mean curvature flow under symmetry assumptions, extending the theoretical understanding of these solitons.

Findings

Explicit solutions for graphical translators are obtained.

Properties of these translators are characterized.

Existence and uniqueness results are established.

Abstract

In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features.