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Geometric inequalities outside a convex set in a Riemannian manifold

Authors
Seo, K
Issue Date
2007
Publisher
KINOKUNIYA CO LTD
Citation
JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY, v.47, pp.657 - 664
Journal Title
JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY
Volume
47
Start Page
657
End Page
664
URI
https://scholarworks.sookmyung.ac.kr/handle/2020.sw.sookmyung/148486
Abstract
Let M be an n-dimensional complete simply connected Riemannian manifold with nonpositive sectional curvature for n = 2, 3 and 4. We prove the following Faber-Krahn type inequality for the first eigenvalue lambda(1) of the mixed boundary problem. A domain Omega outside a closed convex subset C in M satisfies lambda(1)(Omega) >= lambda(1)(Omega*) with equality if and only if Omega is isometric to the half ball Omega* in R-n, whose volume is equal to that of Omega. We also prove the Sobolev type inequality outside a closed convex set C in M.
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