Translating solitons of translation and homothetical types

TL;DR

This paper characterizes translating solitons in Euclidean and Lorentz-Minkowski spaces, showing that certain decompositions imply the soliton must be a plane or grim reaper, thus advancing understanding of their geometric structure.

Contribution

It proves new conditions under which translating solitons are planar or grim reapers, extending previous results to product structures and Lorentz-Minkowski space.

Findings

If a translating soliton is a sum of two curves with one planar, the other is also planar.

Such solitons are necessarily planes or grim reapers.

Results extend to solitons expressed as products of functions in Lorentz-Minkowski space.

Abstract

We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate translating solitons that can be locally written as the product of two functions of one variable. We extend the results in Lorentz-Minkowski space.