Spacelike Self-Similar Shrinking Solutions of the Mean Curvature Flow in Pseudo-Euclidean Spaces

TL;DR

This paper classifies certain spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-Euclidean spaces, extending known Euclidean results and excluding their existence in specific cases.

Contribution

It provides a classification of spacelike self-similar shrinking solutions in pseudo-Euclidean spaces under specific geometric conditions, generalizing Euclidean classifications.

Findings

Classification of solutions under given conditions

Exclusion of solutions in certain cases

Extension of Euclidean results to pseudo-Euclidean spaces

Abstract

I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cases. The classification is analogous to the existing classifications in the euclidean case of Huisken and Smoczyk.