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The isoperimetric problem on surfaces. (English) Zbl 1003.52011
The paper begins by displaying distinct approaches to the proof of the isoperimetric problem in the Euclidean plane. It proceeds to deal with the generalization of this problem to several surfaces of revolution and to the paraboloid, and the authors find a unique solution in each case. The paper finishes by considering an analogous problem in hyperbolic manifolds.
It should be pointed out that, throughout the paper, results and ideas are explained clearly and concisely, with good references to literature of the subject.

MSC:
52B60 Isoperimetric problems for polytopes
53A05 Surfaces in Euclidean and related spaces
49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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