  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.

Differential Geometry