Compactness and Rigidity of $\hat{I}»$-Surfaces

Abstract

In this paper we develop the compactness theorem for $\hat{I}»$-surface in ${\mathbb{R}}^3$ with uniform $\hat{I}»$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for self-shrinkers in ${\mathbb{R}}^3$. As an application of this compactness theorem, we prove a rigidity theorem for convex $\hat{I}»$-surfaces.

19 pages. Correct a mistake pointing out by Jonathan Zhu. Comments are welcomed!