Singular Support of Minimizers of the Causal Variational Principle on the Sphere

TL;DR

This paper investigates the structure of minimizing measures for the causal variational principle on the sphere, revealing geometric and dimensional properties depending on the parameter .

Contribution

It proves that for >, the support lies on finitely many real analytic curves intersecting finitely, and for >, the support's Hausdorff dimension is at most 6/7.

Findings

Support consists of finitely many real analytic curves for >.

Support has Hausdorff dimension at most 6/7 for >.

Supports intersect at finitely many points.

Abstract

The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $\tau>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $\tau>\sqrt{6}$, the support is proven to have Hausdorff dimension at most $6/7$.