Local Minimizers of the Anisotropic Isoperimetric Problem on Closed Manifolds

TL;DR

This paper characterizes local minimizers of the anisotropic isoperimetric problem on closed manifolds, showing they are close to tangent Wulff shapes and geodesically convex, especially for small volumes.

Contribution

It provides a quantitative description of local minimizers as small perturbations of tangent Wulff shapes on closed Riemannian manifolds.

Findings

Local minimizers are geodesically convex.

They are small smooth perturbations of tangent Wulff shapes.

Results are quantitative in terms of volume.

Abstract

Local minimizers for the anisotropic isoperimetric problem in the small-volume regime on closed Riemannian manifolds are shown to be geodesically convex and small smooth perturbations of tangent Wulff shapes, quantitatively in terms of the volume.