Ricci Flow of Anosov Metrics on Surfaces

Definitions and mathematical statement

Let $M$ be a closed smooth manifold, and let $\operatorname{Met}^{\infty}(M)$ denote its space of smooth Riemannian metrics. For $g\in\operatorname{Met}^{\infty}(M)$, write $S_gM$ for the unit tangent bundle and $\varphi_g^t$ for the geodesic flow. This flow is Anosov if there is a continuous invariant splitting \[ T(S_gM)=E^s\oplus{\mathbb{R}} X_g\oplus E^u, \] where $X_g$ generates the flow, and constants $C,\lambda>0$ such that, for every $t\geq0$, \[ \|D\varphi_g^t(v)\|\leq Ce^{-\lambda t}\|v\|\quad(v\in E^s),\qquad \|D\varphi_g^{-t}(w)\|\leq Ce^{-\lambda t}\|w\|\quad(w\in E^u). \] Here the norms may be taken with respect to a fixed auxiliary Riemannian metric on $S_gM$. Define \[ \operatorname{Met}_{\mathrm{Anosov}}(M) =\{g\in\operatorname{Met}^{\infty}(M):\varphi_g^t\text{ is Anosov}\}. \] Equip this space with the subspace topology induced by the smooth $C^{\infty}$ topology; this is an explicit convention because the supplied record does not specify a topology.

The original record asks whether $\operatorname{Met}_{\mathrm{Anosov}}(M)$ is contractible. Explicitly, does there exist $g_*\in\operatorname{Met}_{\mathrm{Anosov}}(M)$ and a continuous map \[ H:\operatorname{Met}_{\mathrm{Anosov}}(M)\times[0,1] \longrightarrow\operatorname{Met}_{\mathrm{Anosov}}(M) \] such that $H(g,0)=g$ and $H(g,1)=g_*$ for every $g$ in this space? The record does not specify the intended range of closed manifolds or explicitly assume that this space is nonempty. Thus a universal claim over all closed manifolds is not asserted here.

The surface question retained in the cleaned statement is the following. Let $S$ be a closed surface of genus $h\geq2$, and let $g_0\in\operatorname{Met}_{\mathrm{Anosov}}(S)$. Here genus is interpreted in the usual orientable-surface sense; orientability is not separately stated in the record. Does Ricci flow starting from $g_0$ reach a metric with strictly negative Gaussian curvature everywhere at some finite positive time?

More precisely, the normalized interpretation used in the supplied literature assessment is \[ \partial_tg(t)=-2\left(K_{g(t)}-\overline K_{g(t)}\right)g(t),\qquad g(0)=g_0, \qquad \overline K_g=\frac{\int_S K_g\,\mathrm dA_g}{\int_S\mathrm dA_g}. \] Writing $[0,T_{\max})$ for the maximal forward interval of a smooth solution, the question is whether, for every such $S$ and $g_0$, \[ \exists\,t_0\in(0,T_{\max})\quad \forall\,x\in S,\qquad K_{g(t_0)}(x)<0. \] The statement itself does not specify normalization; the unnormalized interpretation instead uses $\partial_tg=-2\operatorname{Ric}_g=-2K_g g$, with the same finite-time curvature question. No additional requirement that the intervening metrics remain Anosov is explicit in the supplied surface statement. Such a requirement would be a separate formulation relevant to paths inside $\operatorname{Met}_{\mathrm{Anosov}}(S)$.

The supplied dated assessment says that normalized Ricci flow answers the surface question through convergence to constant negative curvature, reports path connectivity for an unspecified class of Anosov metrics, and identifies contractibility of the full space as unresolved. These are supplied curation claims, not independently verified conclusions of this draft. In particular, the surface question and the broader contractibility question must be distinguished despite the record's overall open label.

Short English statement

The surface question asks whether Ricci flow from every Anosov metric on a closed surface of genus at least $2$ reaches strictly negative curvature everywhere in finite time. The original record also asks whether the space of Anosov metrics on a closed manifold is contractible. The supplied dated assessment treats the surface question as answered but the broader question as unresolved; flow normalization, topology, and the intended manifold class require review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003369$, title "Ricci Flow of Anosov Metrics on Surfaces," exact source code catalogue entry. The record gives proposed year $2017$ and names no individual proposer.
• Supplied report citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The title's year $2018$ differs from the DOI's $2017$ component and the proposed year; this discrepancy is retained for review.
• Supplied literature source: Path Connectivity of Anosov Metrics on Surfaces. https://arxiv.org/abs/2601.08656. The supplied record does not give authors or the hypotheses defining the class covered by its path-connectivity assessment.
• Supplied literature source: Oberwolfach Report 3/2017. https://ems.press/content/serial-article-files/46667.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified here. It labels the status open, states that normalized Ricci flow answers the surface example by convergence to constant negative curvature, reports path connectivity for a class of Anosov metrics, and describes contractibility of the full space as unresolved. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official passage was checked and rewritten while preserving its mathematical content. These are supplied verification claims; this draft does not claim to have consulted the report or independently verified them.