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

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.

Revised version, 33 pages; 7 figures added; more details in Section 5.2 and Lemma A.2 added; references updated