Relative isoperimetric inequality for minimal surfaces 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 non-posit constant K. Assume that S 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. If 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, then we prove a sharp relative isoperimetric inequality 2 pi Area(Sigma) <= Length(partial derivative Sigma similar to partial derivative C)(2) + KArea(Sigma)(2), where equality holds if and only if S is a geodesic half disk with constant Gaussian curvature K. We also prove the relative isoperimetric inequalities for minimal submanifolds outside a closed convex set in a higher-dimensional Riemannian manifold.