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An isoperimetric inequality on conformally hyperbolic manifolds. (English. Russian original) Zbl 1075.53027

Sb. Math. 194, No. 4, 495-513 (2003); translation from Mat. Sb. 194, No. 4, 29-48 (2003).
An inequality \[ P(v(D)) \leq s(\partial D) \] between the volume of a domain \(D\) in a Riemannian manifold \(M\) and the area \(s(\partial D)\) of its boundary is called an isoperimetric inequality, and the function \(P\) (a non-trivial and rather a maximal) is called an isoperimetric function of \(M\). The author develops the methods of a joint paper with V. A . Zorich further and proves that on each Riemannian manifold of conformally hyperbolic type a conformal change of the original Riemannian metric (i.e. the change \(g \to \tilde g= \lambda^2 g\), where \(\lambda\) is a regular positive function on \(M\)) takes the isoperimetric inequality to the same form \(v(D) \leq s(\partial D)\) as in the standard Lobachevskii space.

MSC:

53C20 Global Riemannian geometry, including pinching
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
31C15 Potentials and capacities on other spaces
31C45 Other generalizations (nonlinear potential theory, etc.)
53A30 Conformal differential geometry (MSC2010)
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