Abstract
We prove that the maximal isoperimetric function on a Riemannian manifold of conformally hyperbolic type can be reduced to the linear canonical form P(x) = x by a conformal change of the Riemannian metric. In other words, the isoperimetric inequality \(P\left( {V\left( D \right)} \right) \leqslant {\text{S}}\left( {\partial {\text{D}}} \right)\), relating the volume V(D) of a domain D to the area \({\text{S}}\left( {\partial {\text{D}}} \right)\) of its boundary, can be reduced to the form \(\left( {V\left( D \right)} \right) \leqslant {\text{S}}\left( {\partial {\text{D}}} \right)\), known for the Lobachevskii hyperbolic space.
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Zorich, V.A., Kesel'man, V.M. The Isoperimetric Inequality on Manifolds of Conformally Hyperbolic Type. Functional Analysis and Its Applications 35, 90–99 (2001). https://doi.org/10.1023/A:1017523114581
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DOI: https://doi.org/10.1023/A:1017523114581