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 nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point \(p \in \partial\Sigma\cap \partial C\), then we prove a sharp relative isoperimetric inequality $$2\pi{\rm Area}(\Sigma) \leq {\rm Length}(\partial\Sigma \sim \partial C)^2 + K{\rm Area}(\Sigma)^2,$$ where equality holds if and only if Σ 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.