Complete self-shrinkers of the mean curvature flow

Abstract

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for \(\mathcal {L}\)-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on polynomial volume growth, which is assumed in Cao and Li [5]. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on polynomial volume growth. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.