论文标题
恒定加权平均曲率超曲面的刚度定理
Rigidity theorems for constant weighted mean curvature hypersurfaces
论文作者
论文摘要
在本文中，我们研究了Hypersurfaces $σ\ subset \ Mathbb {r}^{n+1} $，具有恒定加权平均曲率。最近，Wei-Peng证明了CWMC Hyperfaces的刚性定理，该定理将Le-Sesum分类定理概括为自我脱落器。更具体地说，他们表明，具有多项式体积增长的完整CWMC超表面，第二个基本形式的有界规范，并且满足$ | a | a |^2h（h-λ）\ leq h^2/2 $必须是超平面或广义圆柱体。我们通过删除第二种基本形式规范的界限来概括这一结果。此外，我们证明，在某些条件下，如果反向不等式成立，则超出表面必须是超平面或广义圆柱体。作为本文证明的结果之一的应用，我们将获得由本文作者获得的另一个版本的分类定理，也就是说，在某些条件下，具有$ H \ geq 0 $的完整CWMC Hypersurface必须是超平面或普遍的圆柱体。
In this article, we study hypersurfaces $Σ\subset \mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for self-shrinker. More specifically, they showed that a complete CWMC hypersurface with polynomial volume growth, bounded norm of the second fundamental form and that satisfies $|A|^2H(H-λ)\leq H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions if the reverse inequality holds then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete CWMC hypersurface with $H\geq 0$ must either be a hyperplane or a generalized cylinder.
