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arXiv · 2503.14132

A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces

Enrico Pasqualetto·Tapio Rajala
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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.

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Enrico Pasqualetto, Tapio Rajala. 2025-03-18. A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces. https://arxiv.org/abs/2503.14132

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