Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons

TL;DR

This paper investigates the spectral and geometric properties of self-shrinkers immersed in gradient shrinking Ricci solitons, revealing restrictions on their possible configurations and domains.

Contribution

It establishes new spectral properties of the drifted Laplacian and derives geometric constraints for self-shrinkers in shrinking Ricci solitons.

Findings

Certain domains in the ambient space cannot contain self-shrinkers.

Spectral properties of the drifted Laplacian are characterized.

Geometric restrictions on self-shrinkers are derived.

Abstract

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.