A surface with discontinuous isoperimetric profile and expander manifolds

TL;DR

This paper constructs specific expander manifolds to demonstrate the existence of a 2D Riemannian surface with a discontinuous isoperimetric profile, and extends these ideas to 3-spheres with large separating surfaces.

Contribution

It introduces the concept of expander manifolds in geometry and uses them to answer a longstanding question about isoperimetric profiles and surface separation in Riemannian manifolds.

Findings

Existence of a 2D Riemannian manifold with discontinuous isoperimetric profile.

Construction of 3-spheres where separating surfaces must have arbitrarily large area.

Application of expander manifolds to geometric analysis problems.

Abstract

We construct sequences of `expander manifolds' and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any $\epsilon , M>0$ there is a Riemannian 3-sphere $S$ of volume 1, such that any (not necessarily connected) surface separating $S$ in two regions of volume greater than $\epsilon $, has area greater than $M$.