Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension

TL;DR

This paper constructs a counterexample in higher codimension showing that the relationship between isoperimetric inequalities and minimal submanifolds does not extend beyond codimension two, challenging previous assumptions.

Contribution

It provides the first known counterexample demonstrating the failure of a White-type implication in codimension greater than two.

Findings

Counterexample in codimension > 2

Existence of a Riemannian metric with convex boundary and no closed geodesics

Presence of a complete nonconstant geodesic

Abstract

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four-dimensional ball B with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.