Abstract
This paper investigates the existence of an area (or Dirichlet integral) minimizing parametric surface in a hyperbolic 3-manifold subject to a volume constraint. The existence of a minimizing surface is proved, assuming some conditions on the prescribed free homotopy class. This result implies a non-existence result of minimizing surfaces of prescribed mean curvature. A criterion for the existence of surfaces of prescribed mean curvature, which turns out to be optimal in view of the non-existence result, is also obtained.
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Toda, M. On Minimizing Problems with a Volume Constraint in Hyperbolic 3-Manifolds. Annals of Global Analysis and Geometry 17, 19–42 (1999). https://doi.org/10.1023/A:1006576216626
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DOI: https://doi.org/10.1023/A:1006576216626