PRESSURE GAPS, GEOMETRIC POTENTIALS, AND NONPOSITIVELY CURVED MANIFOLDS

The paper proves a general pressure-gap criterion (Theorem A) for rank-1 manifolds with a codimension-1 totally geodesic flat torus using a shadowing map Π_{t,R}, quantitative time–angle estimates (3.4)–(3.5), a key integral lower bound (Proposition 3.4/Corollary 3.5), and an abstract specification-based construction (Proposition 7.2). This yields P(Sing,ϕ) < P(ϕ) under the stated decay (1.1) without appealing to the geometric potential ϕ^u and without assuming h_top(Sing)=0 . By contrast, the candidate solution hinges on two problematic steps: (i) it asserts P(Sing,ϕ)=C0 by claiming all measures on Sing have zero entropy; the paper only uses P(Sing,ϕ)=C0+h_top(Sing) (when ϕ|Sing≡C0) and does not assume h_top(Sing)=0, although that may hold in many examples ; and (ii) it requires a two-sided estimate −ϕ^u≈|x|^{m/2}+|φ|^{m/(m+2)} from normal curvature bounds alone, whereas the paper proves such an equivalence (Theorem C) under a Ricci curvature hypothesis and no flat strips, which is stronger than the assumptions of Theorem A . Even granting (ii), the argument further concludes P(ϕ)≥C0+P(qϕ^u) by comparing only on a neighborhood of Sing; this does not imply P(ϕ)>P(Sing,ϕ) unless one also proves h_top(Sing)=0. The paper’s proof avoids these gaps by a different mechanism (shadowing and specification), so the statement is correct as proved in the paper, while the model’s proof is incomplete.