Relative isoperimetric inequalities for minimal submanifolds outside a convex set

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 Mp(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.