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Isoperimetric regions in the hyperbolic plane between parallel horocycles. (English) Zbl 1266.53038

The reviewed paper is devoted to the classical isoperimetric problem in the hyperbolic plane. Let \(\mathcal F\) be the region inside two parallel horocycles of \(\mathbb R^2_+\), represented by two horizontal Euclidean straight lines. The author considers the following problem: Fix an area value and study the domains \(\Omega\subset\mathcal F\) with the prescribed area which have minimal free boundary perimeter, but not counting its part of the boundary contained in the horocycles. He gives a detailed and complete classification of the isoperimetric solutions through isoperimetric inequalities.

MSC:

53C20 Global Riemannian geometry, including pinching
53A35 Non-Euclidean differential geometry
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Full Text: Euclid

References:

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