New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

TL;DR

This paper explores translating solitons in Generalised Robertson-Walker spacetimes, classifying new examples and extending classical Grim Reaper solutions within Lorentzian geometry.

Contribution

It introduces the concept of translators in these spacetimes, identifies specific warping functions allowing their existence, and classifies analogues of classical solutions.

Findings

Identified three families of warping functions for translators.

Classified analogues of Grim Reapers in this setting.

Extended the understanding of mean curvature flow in Lorentzian manifolds.

Abstract

Translators can be regarded as submanifolds which satisfy the mean curvature flow equation when evolving by translations along a distinguished vector field of the ambient space. We study translators in Generalised Robertson-Walker spacetimes, due to their importance as Lorentzian manifolds, and because they admit a natural conformal Killing timelike vector field carrying substantial geometric information, which will play the role of this translating vector field. We identify three one-parameter families of warping functions for which these objects exist. As a first example of this notion of translator, we classify the analogues of the classical Grim Reapers within this context.