Symmetry of Stationary Hypersurfaces in Hyperbolic Space
Symmetry of Stationary Hypersurfaces in Hyperbolic Space
复制标题
DOI：
10.1007/s10711-006-9048-1
复制
发表时间：
2006-04
影响因子：
0.5
通讯作者：
R. López
中科院分区：
文献类型：
--
作者：
R. López
我们将规定量的液体沉积在双曲空间的脐超曲面 Π 上。在存在指向 Π 的均匀重力矢量场的情况下，我们寻求机械系统平衡状态下这种液滴的形状。然后，液体-空气界面通过超曲面建模，条件是其平均曲率是距 Π 的距离的函数，并且与 Π 沿其边界形成的角度是恒定的。我们证明超曲面相对于与 Π 正交的测地线是旋转对称的。我们将此结果扩展到其他配置，例如，两个脐带超曲面之间的液桥。最后，我们得到的结果表明，在平均曲率的某些假设下，嵌入超曲面从其边界继承了一定的对称性。
We deposit a prescribed amount of liquid on an umbilical hypersurface Π of the hyperbolic space. Under the presence of a uniform gravity vector field directed towards Π, we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface in then modeled by a hypersurface under the condition that its mean curvature is a function of the distance from Π, together with the fact that the angle that makes with Π along its boundary is constant. We show that the hypersurface is rotational symmetric with respect to a geodesic orthogonal to Π. We extend this result to other configurations, for example, liquid bridges trapped between two umbilical hypersurfaces. Finally, we obtain a result which says that, under some assumptions on the mean curvature, an embedded hypersurface inherits a certain symmetry from its boundary.