Translating solitons to higher order mean curvature flows in Riemannian products

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.