Pinching theorems for self-shrinkers of higher codimension (Q6645890)

This paper considers pinching phenomena of the tracefree second fundamental form of complete self-shrinkers of higher codimension. A self-shrinker is a time slice of a self-similar solution of the mean curvature flow, which shrinks as time increases. It is well-known that self-shrinkers play an important role in the study of mean curvature flow for they describe the singularity models of the mean curvature flow and they arise as tangent flows of mean curvature flow at singularities.\N\NFirst, under the assumption that the mean curvature is nonzero everywhere and the self-shrinker is of polynomial volume growth, the authors prove that if the tracefree second fundamental form \(\tilde{A}\) satisfies \(||\tilde{A}||_n < C(n)\) for a positive constant \(C(n)\) depending only on the dimension n of the self-shrinker, then it is isometric to the sphere \(S_n( \sqrt{2n})\).\N\NMoreover, assuming that the mean curvature vector \(H\) of the self-shrinker satisfies \(\sup |H| < \sqrt{\frac \pi 2}\) and \(\tilde{A}\) satisfies \(||\tilde{A}||_n < D(n, \sup |H|)\) for a positive constant \(D(n,\sup |H|)\) depending on \(n\) and \(\sup |H|\), the authors show that the self-shrinker is isometric to the Euclidean space \(\mathbb{R}^n\).\N\NLastly, some rigidity theorems for self-shrinkers satisfying pointwise curvature pinching conditions on \(|\tilde{A}|^2 \) are also proved.