On a classification theorem for self-shrinkers

TL;DR

This paper extends classification results for self-shrinkers in mean curvature flow by replacing volume growth assumptions with weighted L^2 conditions, broadening the understanding of their geometric properties.

Contribution

It introduces a new classification theorem for self-shrinkers using weighted L^2 conditions, generalizing previous polynomial volume growth assumptions.

Findings

Extended classification results for self-shrinkers.

Recovered and extended recent classification and gap results.

Utilized weighted manifold perspective for broader applicability.

Abstract

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the viewpoint of weighted manifolds and permits also to recover and to extend some others recent classification and gap results for self-shrinkers.