The Isoperimetric Flow and Synthetic Scalar Curvature for Causal Variational Principles

Abstract

An isoperimetric flow is introduced in the setting of causal variational principles. Describing the whole space by the support of a minimizing measure, we define an area functional on subsets by a surface layer integral. Mean curvature is defined via the first variation of this area functional. Using the method of minimizing movement, the isoperimetric flow is constructed as a Lipschitz curve of measures with prescribed volume change. As illustrating examples, we introduce and analyze ``smeared variants'' of the mean curvature flow in Euclidean space and on a Riemannian manifold. In these examples, it is shown that our area functional coincides with the usual area up to specified curvature corrections. As an application of our general results, by analyzing the volume and area of small isoperimetric balls, we introduce notions of local dimension and synthetic scalar curvature.

arXiv comment33 pages, LaTeX, 3 figures