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 $$\epsilon , M>0$$ ϵ , 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 $$\epsilon $$ ϵ , has area greater than M .

中文翻译：

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具有不连续等周轮廓和扩展歧管的表面

我们构建了“扩展流形”的序列，并使用它们来证明存在具有不连续等周剖面的完整连接的二维黎曼流形，回答了 Nardulli 和 Pansu 的问题。使用维度 3 中的扩展流形，我们表明对于任何 $$\epsilon ， M>0$$ ϵ ，M > 0 存在体积为 1 的黎曼 3 球体 S，使得任何（不一定连接）表面将 S 分隔开体积大于 $$\epsilon $$ ϵ 的两个区域的面积大于 M 。