Graphical translating solitons for the mean curvature flow and isoparametric functions

TL;DR

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

Contribution

It provides a new classification of translating solitons related to isoparametric functions and analyzes their shape based on solutions to associated differential equations.

Findings

Classification of translating solitons based on isoparametric functions.

Analysis of the shape of solutions to the governing ODEs.

Results for cases with 1, 2, or 3 principal curvatures.

Abstract

In this paper, we consider a translating soliton for the mean curvature flow starting from a graph of a function on a domain in a unit sphere which is constant along each leaf of isoparametric foliation. First, we show that such a function is given as a composition of an isoparametric function on the 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. Finally, we investigate a domain of the function which is given as a composition of the isoparametric function and the solution of the ordinary differential equation in the case where the number of distinct principal curvatures of the isoparametric hypersurface defined by the regular level set for the isoparametric function is 1, 2, or 3.