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 \emph{polynomial volume growth}, which is assumed in Cao and Li. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on \emph{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.

10 pages, a revised version

References in corpus (6)

• Generic mean curvature flow I; generic singularities
• Volume growth, eigenvalue and compactness for self-shrinkers
• The rigidity theorems of self shrinkers
• Classification and rigidity of self-shrinkers in the Mean curvature flow
• Volume estimate about shrinkers
• Estimates for eigenvalues of $\mathfrak{L}$ operator on Self-Shrinkers

Cited by in corpus (2)

• The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow
• On a classification theorem for self-shrinkers