Some classifications of 2-dimensional self-shrinkers

Abstract

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).