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Relative isoperimetric inequalities for minimal submanifolds outside a convex set
Author(s) -
Seo Keomkyo
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100078
Subject(s) - isoperimetric inequality , mathematics , sectional curvature , riemannian manifold , convex set , minimal surface , pure mathematics , gaussian curvature , geodesic , constant curvature , combinatorics , regular polygon , constant (computer programming) , manifold (fluid mechanics) , scalar curvature , ricci curvature , mathematical analysis , convex body , bounded function , simply connected space , curvature , convex hull , geometry , convex optimization , mechanical engineering , computer science , engineering , programming language
Abstract Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K . Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂ C along ∂Σ∩∂ C and ∂Σ ∼ ∂ C is radially connected from a point p ∈ ∂Σ∩∂ C . We introduce a modified volume M p (Σ) of Σ and obtain a sharp isoperimetric inequality\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ 2\pi M_p (\Sigma ) \le {\rm Length}(\partial \Sigma \sim \partial C)^2, $$ \end{document} where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K . We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.