Relative isoperimetric inequalities for minimal submanifolds outside a convex set

초록

Let C be a closed convex set in a complete simply connected Riemannian manifoldM with sectional curvature bounded above by a positive constant K. Assume that Sigma is a compact minimal surface outside C such that Sigma is orthogonal to partial derivative C along partial derivative Sigma boolean AND partial derivative C and partial derivative Sigma similar to partial derivative C is radially connected from a point p is an element of partial derivative Sigma boolean AND partial derivative C. We introduce a modified volume M-p(Sigma) of Sigma and obtain a sharp isoperimetric inequality 2 pi M-p (Sigma) <= Length(partial derivative Sigma similar to partial derivative C)(2), where equality holds if and only if Sigma 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. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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