Translating solitons from semi-Riemannian foliations

TL;DR

This paper studies translating solitons in semi-Riemannian product manifolds, providing criteria for lifting and projecting solitons via submersions and constructing explicit examples through differential equations.

Contribution

It introduces criteria for lifting and projecting translating solitons in semi-Riemannian foliated manifolds, enabling systematic construction of explicit soliton examples.

Findings

Criteria for lifting and projecting solitons in semi-Riemannian submersions

Explicit construction of solitons using differential equations

Unified approach to known and new soliton examples

Abstract

We recall the notion of (vertical) translating solitons in a product of a semi-Riemannian manifold $(M,g)$ and the real line. Mainly, we restrict our attention to those which are the graph of a smooth function. When dealing with submersions, we show a criteria to lift (or project) translating solitons from the base manifold to the total space (or viceversa). In particular, manifolds foliated by codimension 1 orbits of a Lie group action give rise to such solitons, up to solving a first-order ordinary differential equation. This gives us explicit criteria under which the graph of a function is a soliton, and we employ them to construct many examples of solitons, both new and old, in a unified way.