Higher codimension isoperimetric problems

Abstract

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutions of this higher codimension CMC prob-lem; these concentrate near the critical points of a certain curvature function.