Complete self-shrinkers confined into some regions of the space

TL;DR

This paper investigates the geometric and spectral properties of complete non-compact self-shrinkers, establishing conditions under which they are compact and exploring their intersections with hyperplanes, contributing to the understanding of their volume growth.

Contribution

It provides new restrictions on self-shrinkers' compactness, analyzes their intersection behavior with hyperplanes, and offers spectral insights related to the drifted Laplacian, advancing the study of their volume growth.

Findings

Complete non-compact self-shrinkers can be forced to be compact under certain conditions.

Self-shrinkers intersect hyperplanes transversally, with implications for their structure.

Spectral properties of the drifted Laplacian are derived when intersections are compact.

Abstract

We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H. D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented.