Relative isoperimetric inequality for minimal submanifolds in space forms

Abstract

Let C be a closed convex set in Sm or Hm. Assume that ∑ is an n-dimensional compact minimal submanifold outside C such that ∑ is orthogonal to ∂C along ∂∑ ∩ ∂C and ∂∑ lies on a geodesic sphere centered at a fixed point p ∈ ∂∑ ∩ ∂C and that r is the distance in Sm or Hm from p. We make use of a modified volume Mp(∑) of ∑ and obtain a sharp relative isoperimetric inequality ½nnωnMp(∑)n-1 ≤ Vol(∂∑ ~ ∂C)n,where ωn is the volume of a unit ball in Rn. Equality holds if and only if ∑ is a totally geodesic half ball centered at p.