Copyright © 2010 Juncheol Pyo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Σ be a domain on an m-dimensional minimal submanifold in the outside of a convex set C in 𝕊n or ℍn. The modified volume M(Σ) is introduced by Choe and Gulliver (1992) and we prove a sharp modified relative isoperimetric inequality for the domain Σ, (1/2)mmωmM(Σ)m-1≤Volume(∂Σ-∂C)m, where ωm is the volume of the unit ball of ℝm. For any domain Σ on a minimal surface in the outside convex set C in an n-dimensional Riemannian manifold, we prove a weak relative isoperimetric inequality πArea(Σ)≤Length(∂Σ-∂C)2+KArea(Σ)2, where K is an upper bound of sectional curvature of the Riemannian manifold.