Pressure gaps, geometric potentials, and nonpositively curved manifolds

In this paper, we derive a general pressure gap criterion for closed rank 1 manifolds with singular sets characterized by codimension 1 totally geodesic flat subtori. As an application, we demonstrate that under specific curvature constraints, potentials that decay faster than geometric potentials (towards the singular set) exhibit pressure gaps and lack phase transitions. Additionally, we prove that geometric potentials are Hölder continuous near singular sets.