Ruelle Zeta Vanishing for Generic Negatively Curved Metrics

Definitions and mathematical statement

Let $\Sigma$ be a closed, connected, oriented three-manifold admitting a negatively curved Riemannian metric, and let $\mathcal G_-(\Sigma)$ denote its space of negatively curved Riemannian metrics. Here negatively curved means that every sectional curvature is strictly negative. Write \[ b_1(\Sigma)=\dim_{{\mathbb{R}}}H^1(\Sigma;{\mathbb{R}}). \] For $g\in\mathcal G_-(\Sigma)$, consider the geodesic flow on the unit tangent bundle $S_g\Sigma$, and let $\zeta_g(s)$ denote its Ruelle zeta function with the convention intended by the source.

The source does not specify the zeta normalization. For example, a product convention is \[ \prod_{\gamma\in\mathcal P_g}\left(1-e^{-s\ell_g(\gamma)}\right), \qquad \operatorname{Re}s\text{ sufficiently large}, \] where $\mathcal P_g$ consists of primitive periodic trajectories of the geodesic flow, counted up to time translation, and $\ell_g(\gamma)$ is their period. The reciprocal convention reverses the signed order at zero. This example does not identify which convention the supplied source intends.

For a nonzero meromorphic germ $\zeta_g$ at zero, define its signed order $m_g(0)\in{\mathbb{Z}}$ by \[ \zeta_g(s)=s^{m_g(0)}h_g(s),\qquad h_g\text{ holomorphic near }0,\quad h_g(0)\ne0. \] Positive order denotes a zero, negative order a pole, and order zero a regular nonzero value.

Conjectural target from the supplied record. For every such $\Sigma$, is it true that, for a generic $g\in\mathcal G_-(\Sigma)$, \[ m_g(0)=4-b_1(\Sigma)? \] The record leaves the regularity and topology on the metric space, the meaning of genericity, and the zeta normalization unspecified. These must be fixed before this becomes a fully precise assertion. Its phrase "order of vanishing" also requires clarification when $4-b_1(\Sigma)<0$; the signed-order interpretation above allows poles.

The supplied dated curation labels the problem partially solved and reports the asserted value for generic conformal perturbations of compact hyperbolic three-manifolds. It explicitly says that this does not establish the claim for generic negatively curved metrics in full generality. This is a supplied literature assessment, not an independently verified result here.

Short English statement

The conjecture asks whether a generic negatively curved metric on each closed, connected, oriented three-manifold $\Sigma$ has Ruelle zeta order $4-b_1(\Sigma)$ at zero. The intended genericity, metric regularity, zeta convention, and treatment of poles need clarification. Supplied dated curation reports partial progress for generic conformal perturbations of compact hyperbolic three-manifolds.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004738$, source problem code catalogue entry, Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2021.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, pp. 1735--1803, DOI 10.4171/owr/2021/33. https://doi.org/10.4171/owr/2021/33. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Ruelle zeta near hyperbolic metrics. https://doi.org/10.1007/s00222-022-01108-x.
• Supplied literature reference labeled "primary preprint": https://arxiv.org/abs/2009.08558.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem partially solved, reports the conjectured value for generic conformal perturbations of compact hyperbolic three-manifolds, and distinguishes this from the full generic negatively curved metric claim. The supplied curation requests independent verification of time-sensitive status and cited claims.