Compactness of self-shrinkers in $\mathbb R^3$ with fixed genus

TL;DR

This paper proves that self-shrinkers in three-dimensional space with fixed genus and bounded entropy form a compact family, leading to uniform bounds on the number of ends based on these parameters.

Contribution

It establishes the compactness of self-shrinkers with fixed genus and bounded entropy in 3, and derives uniform bounds on their number of ends.

Findings

Self-shrinkers with fixed genus and bounded entropy are compact.

Number of ends of such surfaces is uniformly bounded by entropy and genus.

Provides a new understanding of the geometric structure of self-shrinkers.

Abstract

We prove the compactness of self-shrinkers in $\mathbb R^3$ with bounded entropy and fixed genus. As a corollary, we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus.