No breather theorems for the mean curvature flow

TL;DR

This paper investigates the properties of breathers in the mean curvature flow in Euclidean space, proving that under certain conditions, breathers must be solitonic solutions, thus extending understanding of their structure.

Contribution

The paper establishes no breather theorems for noncompact mean curvature flow solutions, showing they must be solitons under specific conditions, which was previously unknown.

Findings

Breathers in noncompact mean curvature flow are necessarily solitons under certain conditions.

The paper proves several no breather theorems in the noncompact setting.

Breathers cannot be non-solitonic solutions in the studied context.

Abstract

In this article we study the breathers of the mean curvature flow in the Euclidean space. A breather is a solution to the mean curvature flow which repeats itself up to isometry and scaling once in a while. We prove several no breather theorems in the noncompact category, that is, under certain conditions, a breather of the mean curvature flow must be a solitonic solution (self-shrinker, self-expander, or translator).