Translating solitons over Cartan-Hadamard manifolds

TL;DR

This paper investigates the existence and behavior of translating solitons for mean curvature flow on Cartan-Hadamard manifolds, revealing how curvature influences their asymptotics and solutions.

Contribution

It establishes existence results for entire graphical translators, analyzes their asymptotic behavior based on curvature, and solves the asymptotic Dirichlet problem under specific conditions.

Findings

Existence of entire graphical translators depends on manifold curvature.

Asymptotic behavior varies with curvature, especially when it tends to minus infinity.

Bounded solutions exist under certain curvature decay conditions.

Abstract

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.