THE RUELLE ZETA FUNCTION AT ZERO FOR NEARLY HYPERBOLIC 3-MANIFOLDS

The uploaded paper proves that for generic conformal perturbations gs = e^{-2sh}g_H of a closed hyperbolic 3-manifold Σ, the order of vanishing of the Ruelle zeta function at zero is n(gs) = 4 − b1(Σ), with m1,0(gs) = b1(Σ) and m2,0(gs) = b1(Σ)+2, and semisimplicity in degrees k=1,2 (Theorem 1 in the paper) , consistent with the abstract . The relation n(X) = m0,0 − m1,0 + m2,0 − m3,0 + m4,0 is explicitly stated (eq. (1.2)) . The paper also details the hyperbolic case degeneracy m1,0=2b1(Σ), m2,0=2b1(Σ)+2 and non-semisimplicity in k=2 , the perturbation framework via Lemma 4.2 (contact case) , and the metric rescaling/conformal variation formula used to transfer the contact perturbation statement to conformal metrics (Section 6.1–6.2) . The key pairing/regularization identity (Theorem 2) linking the pushforward of products of resonant and coresonant forms to a Laplacian expression is also provided and underpins generic nondegeneracy needed for the splitting of Jordan blocks . The candidate solution follows the same architecture: (i) fix the contact manifold via fiberwise dilation, (ii) express n(gs) through alternating multiplicities of zero-resonances, (iii) use cohomological input and duality, (iv) analyze the hyperbolic degeneracies, and (v) prove generic splitting via a first-variation formula and a pushforward identity, finally obtaining m1,0=b1(Σ), m2,0=b1(Σ)+2 and n(gs)=4−b1(Σ). Minor issues: the candidate attributes the main result to a paper including “Delarue” (the uploaded paper’s authors are Cekić, Dyatlov, Küster, Paternain) and overstates a general “two canonical degree‑2 classes” claim for all 5D contact flows; in the paper, a second invariant 2-form is specific to the hyperbolic case (Section 3.1) while dα is always present . These do not affect the core logic or the final conclusion, which matches the paper.