# 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.

## Key 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.