Isoperimetric structure of asymptotically conical manifolds

TL;DR

This paper investigates the isoperimetric properties of asymptotically conical Riemannian manifolds with non-negative Ricci curvature, extending classical results and exploring geometric invariants related to mass and cone angles.

Contribution

It generalizes known isoperimetric and foliation results to asymptotically conical manifolds, providing new insights into their geometric structure and invariants.

Findings

Existence of a canonical foliation by constant mean curvature surfaces at infinity.

Extension of isoperimetric structure results to asymptotically conical manifolds.

Observation on the isoperimetric cone angle as an analogue of the positive mass theorem.

Abstract

We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.-T. Yau on the existence of a canonical foliation by volume preserving stable constant mean curvature surfaces at infinity of asymptotically flat manifolds as well as the results of the second-named author with S. Brendle and J. Metzger on the isoperimetric structure of asymptotically flat manifolds. We also include an observation on the isoperimetric cone angle of such manifolds. This result is a natural analogue of the positive mass theorem in this setting.