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The relative isoperimetric inequality on a conformally parabolic manifold with boundary. (English. Russian original) Zbl 1239.53048

Sb. Math. 202, No. 7, 1043-1058 (2011); translation from Mat. Sb. 202, No. 7, 117-134 (2011).
For an arbitrary noncompact \(n\)-dimensional Riemannian manifold with a boundary of conformally parabolic type the author shows that there exists a conformal change of metric such that a relative isoperimetric inequality of the same form as in the closed \(n\)-dimensional Euclidean half-space holds on the manifold with the new metric. This isoperimetric inequality is asymptotically sharp. The paper gives a full exposition of the theorem presented by the author in [Russ. Math. Surv. 65, No. 2, 384–385 (2010); translation from Usp. Mat. Nauk 65, No. 2, 195–196 (2010; Zbl 1247.53036)]. The result is a natural development of an earlier paper [Sb. Math. 200, No. 1, 1–33 (2009); translation from Mat. Sb. 200, No. 1, 3–36 (2009; Zbl 1168.53008)] by the same author, where he solved a similar problem for an absolute isoperimetric inequality on a manifold with boundary.

MSC:

53C20 Global Riemannian geometry, including pinching
31C45 Other generalizations (nonlinear potential theory, etc.)
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