Translating solitons to higher order mean curvature flows in Riemannian products

TL;DR

This paper establishes existence and classification results for translating solitons in higher order mean curvature flows within warped product manifolds, generalizing known cases like Euclidean and hyperbolic spaces.

Contribution

It introduces new existence and classification results for translating solitons in warped product manifolds, extending previous work to higher order mean curvature flows.

Findings

Existence of bowl-type and catenoid-type translating solitons under mild curvature assumptions.

Classification results for translating solitons in the specified warped product setting.

Asymptotic behavior of solitons linked to the geometry at infinity of the base manifold.

Abstract

In this paper we prove existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows that are invariant by translations in warped product manifolds $\mathbb{P}\times_\chi \mathbb{R}$. Here, $\mathbb P$ is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and $\chi$ is a radial function defined in $\mathbb{P}$. In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor $\mathbb{R}$ in the warped product. This setting encompasses the cases of translating solitons in $\mathbb{R}^{n+1}$, $\mathbb{H}^n \times \mathbb{R}$ and $\mathbb{H}^{n+1}$ studied in recent papers. In particular we prove the existence of families of bowl-type and catenoid-type translating solitons under mild assumptions about the curvature of the warped product. We also describe the asymptotic behavior for those solitons in terms of the geometry at infinity of $\mathbb{P}$. Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.