SPECTRAL GAP FOR SURFACES OF INFINITE VOLUME WITH NEGATIVE CURVATURE

The paper establishes exactly the theorem and bound stated by the model: a uniform resonance-free strip {|λ|>C0, Im λ>−β} for negatively curved even asymptotically hyperbolic (AH) surfaces and the cutoff resolvent estimate ||χ(−Δ−1/4−λ^2)−1χ|| ≤ C |λ|^{−1−C1 min(0,Im λ)} log|λ|; see the main Theorem in the introduction and its proof outline, which proceeds via Vasy’s semiclassical Fredholm reduction, a (weight-free) quantum monodromy/Grushin construction, Eberlein’s 1D trapped set structure, and Vacossin’s 2D open quantum map bounds . The reduction to a compact semiclassical family P(z) and Fredholm/radial estimates is recalled in Proposition 2.3 (and related results) . The quantum monodromy map M(z,h) controlling poles (Proposition 3.1) is constructed precisely as the model describes . The 1D description of the trapped set in 2D negative curvature (Proposition 3.4) is given using Eberlein’s theorem . Vacossin-type estimates on powers of the monodromy map (Propositions 4.1–4.2) produce the spectral gap and lead to the stated resolvent bound after rescaling λ=h^{-1}(1+z) and summing a logarithmic-length series, yielding the log|λ| factor . The only minor discrepancy is attribution of the log-loss: in the paper, the log-factor arises from iterating the monodromy map N≈c log(1/h), not directly from radial point analysis. Otherwise, the model’s solution follows the same structure and conclusions.