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

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 $6/7$.