Abstract
We deposit a prescribed amount of liquid on an umbilical hypersurface Π of the hyperbolic space \(\mathbb{H}^{n+1}\). 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.
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Partially supported by MEC-FEDER Grant no. MTM2004-00109.
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López, R. Symmetry of Stationary Hypersurfaces in Hyperbolic Space. Geom Dedicata 119, 35–47 (2006). https://doi.org/10.1007/s10711-006-9048-1
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DOI: https://doi.org/10.1007/s10711-006-9048-1