Higher codimension relative isoperimetric inequality outside a convex set

Abstract

We consider an isoperimetric inequality for $(m+1)$-dimensional area minimizing submanifolds of arbitrary codimension which lie outside a convex set $\mathcal{K}\subset {\mathbb{R}}^{n+1}$ and are bounded by a submanifold of ${\mathbb{R}}^{n+1}\setminus \mathcal{K}$ and the convex set $\mathcal{K}$. We show that the least value of the isoperimetric ratio is attained for an $(m+1)$-dimensional flat half-disk of ${\mathbb{R}}_{+}^{n+1}$. This extends prior work of Choe, Ghomi, and Ritoré in codimension one and proves a conjecture of Choe in the case of relative area minimizers.

55 pages