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arXiv · 1509.02307

A surface with discontinuous isoperimetric profile and expander manifolds

Panos Papasoglu·Eric Swenson
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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 $M$.

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Panos Papasoglu, Eric Swenson. 2015-09-08. A surface with discontinuous isoperimetric profile and expander manifolds. https://arxiv.org/abs/1509.02307

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