Pressure gaps, geometric potentials, and nonpositively curved manifolds

TL;DR

This paper establishes a pressure gap criterion for certain nonpositively curved manifolds, showing that specific potentials exhibit pressure gaps and no phase transitions, with geometric potentials being H"older continuous near singularities.

Contribution

It introduces a general pressure gap criterion for rank 1 manifolds with singular sets as codimension 1 totally geodesic flats and analyzes potentials decaying towards singular sets.

Findings

Potentials decaying faster than geometric potentials have pressure gaps.

Geometric potentials are H"older continuous near singular sets.

Under curvature constraints, potentials exhibit no phase transitions.

Abstract

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