论文标题
空间形式部分脐带边界的域中的部分过度确定的问题
A partially overdetermined problem in domains with partial umbilical boundary in space forms
论文作者
论文摘要
在本文的第一部分中，我们考虑了空间形式中的部分过度确定的混合边界值问题，并将主要结果推广到\ cite {gx}中，以空间形式的部分脐带边界为一般域。确切地说，我们证明，当且仅当边界的其余部分也是脐带超出表面的一部分时，具有部分脐带边界的域中部分过度确定的问题就可以解决方案。在本文的第二部分中，我们证明了嵌入式超曲面的Heintze-karcher-Ros类型不平等，自由边界位于houshosper上或双曲线空间中的等距性超表面。作为一个应用程序，我们显示了Alexandrov类型定理，用于在这些设置中具有自由边界的恒定平均曲率超曲面。
In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze-Karcher-Ros type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show Alexandrov type theorem for constant mean curvature hypersurfaces with free boundary in these settings.
