Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

TL;DR

This paper classifies graphical translating solitons for the inverse mean curvature flow on spheres, showing they are composed of isoparametric functions and solutions to specific ODEs, with detailed shape analysis.

Contribution

It introduces a novel classification of translating solitons using isoparametric functions and ODE solutions, expanding understanding of inverse mean curvature flow.

Findings

Classification of solitons based on isoparametric functions.

Explicit description of the shape of solutions.

Connection between geometric flow and ODE analysis.

Abstract

In this paper, we consider a translating soliton for the inverse mean curvature flow given as a graph of a function on a domain in a unit sphere whose level sets give isoparametric foliation. First, we show that such function is given as a composition of an isoparametric function on the unit sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons for the inverse mean curvature flow.