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On the uniqueness of isoperimetric solutions and imbedded soap bubbles in non-compact symmetric spaces. I. (English) Zbl 0682.53057
In the case of constant curvature spaces \((E^ n,S^ n(1),H^ n(-1))\) one has the uniqueness of the solution to the isoperimetric problem, namely a round ball. On the other hand, by a theorem of A. D. Alexandrov, imbedded soap bubbles (i.e. closed constant mean curvature surfaces) in \(E^ n,S_+^ n\) or in \(H^ n\) are round spheres, hence congruent if they have same mean curvature. In this paper the authors replace the constant curvature spaces by a simply connected symmetric space with non- positive sectional curvature. They investigate generalizations of the two uniqueness theorems for the special case of products of hyperbolic (or Euclidean) spaces and prove a number of theorems in that direction.
Reviewer: M.Grüter

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q05 Minimal surfaces and optimization
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