Relative isoperimetric inequalities for minimal submanifolds outside a convex set

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.