Pinching Theorems for Self-Shrinkers of Higher Codimension

In this paper, we investigate the pinching phenomena of the tracefree second fundamental form of complete self-shrinkers of higher codimension. Firstly, assuming the mean curvature is nonzero everywhere and the self-shrinker is of polynomial volume growth, we prove that if the tracefree second fundamental form \(\mathring{A}\) $\mathring{A}$ satisfies \(||\mathring{A}||_{n}<C(n)\) $||\mathring{A}{||}_n<C(n)$ for a positive constant C(n) depending only on the dimension n of the self-shrinker, then it is isometric to the sphere \(\mathbb {S}^{n}(\sqrt{2n})\) ${\mathbb{S}}^n(\sqrt{2n})$ . Secondly, we show if the mean curvature vector H of the self-shrinker satisfies \(\sup |H|<\sqrt{\frac{n}{2}}\) $sup|H|<\sqrt{\frac{n}{2}}$ and \(\mathring{A}\) $\mathring{A}$ satisfies \(||\mathring{A}||_{n}<D(n,\sup |H|)\) $||\mathring{A}{||}_n<D(n,sup|H|)$ for a positive constant \(D(n,\sup |H|)\) $D(n,sup|H|)$ depending on n and \(\sup |H|\) $sup|H|$ , then it is isometric to the Euclidean space \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ . We also obtain some rigidity theorems for self-shrinkers satisfying pointwise curvature pinching conditions on \(|\mathring{A}|^{2}\) $|\mathring{A}{|}^2$ .