Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Abstract

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces.

In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov–Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray.

Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile.

Specializing our study to the 2-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary.

The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.