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

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 nonempty. 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 nonbranching spaces, thus in particular to essentially nonbranching spaces and to many Carnot groups and sub‐Riemannian manifolds. As a consequence, every isoperimetric set in an essentially nonbranching space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.