Symmetry of Stationary Hypersurfaces in Hyperbolic Space

Abstract

We deposit a prescribed amount of liquid on an umbilical hypersurfaceof the hyperbolic space H n+1 . Under the presence of a uniform gravity vector field directed towards � , we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface is then modeled by a hypersurface under the condition that its mean curvature is a function of the distance from � , together with the fact that the angle that makes withalong its boundary is constant. We show that the hypersurface is rotational symmetric with respect to a geodesic orthogonal to � .W e 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 assump- tions on the mean curvature, an embedded hypersurface inherits a certain symmetry from its boundary.