Higher codimension isoperimetric problems

TL;DR

This paper introduces a new perspective on higher codimension isoperimetric problems by defining boundary conditions for submanifolds with constant mean curvature and constructs near-spherical solutions concentrating near critical points.

Contribution

It proposes a novel definition for boundary conditions in higher codimension variational problems and constructs approximate solutions concentrating near curvature function critical points.

Findings

Defined boundary conditions with constant mean curvature for higher codimension submanifolds.

Constructed small nearly-spherical solutions concentrating near critical points.

Provided a new approach to 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.