论文标题
平均曲率流的自扩展器的体积特性和刚度
Volume properties and rigidity on self-expanders of mean curvature flow
论文作者
论文摘要
在本文中，我们主要研究在欧几里得空间中浸入的自膨胀性突出，其平均曲率具有一定的线性生长控制。我们讨论了加权体积的体积增长和有限性。我们证明了一些定理，这些定理是通过起源作为自我膨胀者来表征的超级平面。我们估计漂移拉普拉斯曲线底部的上限。我们还为$ l $稳定运算符底部的上限和下限提供了上限，并讨论了一些特殊的自扩展者的$ l $稳定性。 Besides, we prove that the surfaces $Γ\times\mathbb{R}$ with the product metric are the only complete self-expander surfaces immersed in $\mathbb{R}^3$ with constant scalar curvature, where $Γ$ is a complete self-expander curve (properly) immersed in $\mathbb{R}^2$.
In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the $L$-stability operator and discuss the $L$-stability of some special self-expanders. Besides, we prove that the surfaces $Γ\times\mathbb{R}$ with the product metric are the only complete self-expander surfaces immersed in $\mathbb{R}^3$ with constant scalar curvature, where $Γ$ is a complete self-expander curve (properly) immersed in $\mathbb{R}^2$.
