A surface with discontinuous isoperimetric profile and expander manifolds

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 $M$.

15 pages, this paper merged with arXiv:1803.07375 (2018), to appear in Geom. Dedicata