Translating solitons in Riemannian products

TL;DR

This paper investigates translating solitons in Riemannian product manifolds, characterizing their properties, especially in rotationally invariant cases with non-positive curvature, and uses them to derive non-existence results.

Contribution

It introduces a framework for understanding translating solitons in Riemannian products, including their characterization and applications as barriers.

Findings

Translating solitons are minimal for a weighted volume functional.

They naturally appear in a monotonicity formula for mean curvature flow.

Complete classification of rotationally invariant solitons under certain curvature conditions.

Abstract

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in $\mathbb{R} \times P$ can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in $\mathbb{R} \times P$. When $g_0$ is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.