具有形变性质的PI空间中等周集的拓扑正则性
Topological regularity of isoperimetric sets in PI spaces having a deformation property
AI总结：
本文证明了具有形变性质的PI空间中等周集的拓扑正则性，揭示了其开集、边界密度估计及有界性，并推广至塌缩RCD空间，简化了经典欧几里得论证。
摘要：
我们证明了具有适当形变性质的PI空间中等周集的拓扑正则性结果，该性质规定了在用球体进行扰动时集合周长增量的控制。更准确地说，我们证明了等周集是开的，满足边界密度估计，并且在单位球体积的一致下界下是有界的。我们的结果特别适用于可能塌缩的$\mathrm{RCD}(K,N)$空间类。作为推论，在具有欧几里得体积增长的塌缩$\mathrm{RCD}(0,N)$空间上，等周不等式的刚性无需等周集有界性的额外假设即可成立。我们的策略即使在欧几里得环境中也具有意义，因为它简化了一些经典论证。
英文摘要：
We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm{RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.
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