Translators invariant under hyperpolar actions

TL;DR

This paper classifies and analyzes translators for the mean curvature flow on symmetric spaces, focusing on those invariant under hyperpolar actions, with explicit classifications for rank-one cases and investigations for higher rank cases.

Contribution

It provides a classification of translators invariant under hyperpolar actions on symmetric spaces, extending understanding of mean curvature flow in these geometries.

Findings

Classified translators in rank-one symmetric spaces.

Investigated hyperpolar actions of cohomogeneity two in higher rank spaces.

Extended the theory of mean curvature flow invariants in symmetric spaces.

Abstract

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space $G/K$ of compact type which is invariant under a hyperpolar action on $G/K$. First, in the case of $G/K=SO(n+1)/SO(n)$, $SU(n+1)/S(U(1)\times U(n))$, $Sp(n+1)/(Sp(1)\times Sp(n))$ or $F_4/{\rm Spin}(9)$, we classify the shapes of translators in $G/K\times\mathbb R$ given by the graphs of functions on $G/K$ which are invariant under the isotropy action $K\curvearrowright G/K$. Next, in the case where $G/K$ is of higher rank, we investigate translators in $G/K\times\mathbb R$ given by the graphs of functions on $G/K$ which are invariant under a hyperpolar action $H\curvearrowright G/K$ of cohomogeneity two.