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The double cover relative to a convex domain and the relative isoperimetric inequality

Part of: Manifolds Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  09 April 2009

Jaigyoung Choe
Jaigyoung Choe
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea, URL: www.math.snu.ac.kr/~choe, e-mail: choe@math.snu.ac.kr
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Abstract

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We prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

MSC classification

Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting 58E35: Variational inequalities (global problems)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 375 - 382
Copyright
Copyright © Australian Mathematical Society 2006

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