论文标题
HOPF类型定理用于自我脱离器
Hopf type Theorem for Self-Shrinkers
论文作者
论文摘要
在本文中，我们证明了一个二维的自我缩短器，同构对球体的同构，沉浸在三维欧几里得空间中是一个圆形的球体，只要其平均曲率和其位置矢量的规范在其无形的第二基本形式的标准上具有上限。药品构建的示例证明了第二种基本形式的假设是必要的。我们还可以证明，对于径向重量，在R n中具有平行加权平均曲率向量的表面相同的刚度结果。这些结果是在复杂分析中对Cauchy定理进行新的概括的应用，得出的结论是复杂函数相同零或其零，如果满足某些弱的全体形状，则将其隔离。
In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space is a round sphere, provided its mean curvature and the norm of its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in R n with radial weight. These results are applications of a new generalization of Cauchy's Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.
