Title: A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces Show Chinese title

Title: 关于拉普拉斯界、变形性质和度量测度空间中的等周集的注记

Abstract: In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is non-empty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially non-branching ${\sf MCP}(K,N)$ spaces, thus in particular to essentially non-branching ${\sf CD}(K,N)$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially non-branching ${\sf MCP}(K,N)$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed. Show Chinese abstract

Abstract: 在满足适当变形性质的长度PI空间设置中，已知每个等周集都有一个开代表。 在本文中，我们构造了一个长度PI空间（不具有变形性质）的例子，其中等周集没有任何代表，其拓扑内部是非空的。 此外，我们提供了一个变形性质有效的充分条件，该条件涉及从一点出发的平方距离函数的上拉普拉斯界。 我们的结果适用于本质上非分支的${\sf MCP}(K,N)$空间，因此特别适用于本质上非分支的${\sf CD}(K,N)$空间以及许多卡诺群和子黎曼流形。 作为结果，本质上非分支的${\sf MCP}(K,N)$空间中的每个等周集都有一个开代表，在假设单位球体积有统一下界的情况下，该代表也是有界的。