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In this paper, we derive a general pressure gap criterion for closed rank 1manifolds whose singular sets are given by codimension 1 totally geodesic flatsubtori. As an application, we show that under certain curvature constraints,potentials that decay faster than geometric potentials (towards to the singularset) have pressure gaps and have no phase transitions. Along the way, we provethat geometric potentials are H\"older continuous near singular sets.

Pressure gaps, geometric potentials, and nonpositively curved manifolds