Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

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 ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.