Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 758623, 12 pages
Research Article

Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold

Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemungu, Seoul 130-722, South Korea

Received 16 April 2009; Revised 15 December 2009; Accepted 21 January 2010

Academic Editor: Peter. Y. H. Pang

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.


Let Σ be a domain on an m-dimensional minimal submanifold in the outside of a convex set C in 𝕊n or n. 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-1Volume(Σ-C)m, where ωm is the volume of the unit ball of m. 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.