A new pinching theorem for complete self-shrinkers and its generalization

Abstract

In this paper, we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in ℝn+1, and if the squared norm of the second fundamental form of M satisfies \(0\leqslant S - 1\leqslant {1 \over {18}}\), then S ≡ 1 and M is a round sphere or a cylinder. More generally, let M be a complete λ-hypersurface of codimension one with polynomial volume growth in ℝn+1 with λ ≠ 0. Then we prove that there exists a positive constant γ, such that if |λ| ⩽ γ and the squared norm of the second fundamental form of M satisfies \(0\leqslant S - \beta_\lambda\leqslant {1 \over {18}}\), then S ≡ βλ, λ > 0 and M is a cylinder. Here \({\beta _\lambda} = {1 \over 2}(2 + {\lambda ^2} + \left| \lambda \right|\sqrt {{\lambda ^2} + 4})\).