Long-time asymptotics of singular Ricci flow on the remaining non-flat Thurston geometries and a uniform Ricci-flow proof of the generalized Smale conjecture

For a closed 3-manifold $X$ carrying one of Thurston's eight model geometries with model metric $g_X$, the generalized Smale conjecture asserts that the inclusion $\mathrm{Isom}(X,g_X)\hookrightarrow\mathrm{Diff}(X)$ of the isometry group into the diffeomorphism group is a homotopy equivalence. The conjecture is now completely proved for closed orientable prime 3-manifolds: the last case, non-Haken manifolds modeled on the Nil geometry, was settled by Bamler and Kleiner (2023) using singular Ricci flow. What their paper explicitly leaves open is not the conjecture but the program around it: establishing the long-time asymptotics of singular Ricci flow on the remaining non-flat Thurston geometries and thereby extending the uniform Ricci-flow treatment of diffeomorphism groups to all geometrizable 3-manifolds. The problem asks to prove, for manifolds modeled on $\mathbb{H}^2\times\mathbb{R}$, $\widetilde{SL(2,\mathbb{R})}$, or $\mathrm{Solv}$ — and ultimately for arbitrary geometrizable 3-manifolds — that the singular Ricci flow starting from any smooth metric becomes eventually non-singular, satisfies the model curvature pinching (for the $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL(2,\mathbb{R})}$ cases), and converges after rescaling to the model geometry (collapse along Seifert fibers to the hyperbolic base orbifold of curvature $-1/2$ in the pointed Gromov–Hausdorff sense, or closeness to quotients of left-invariant Solv metrics), with the convergence time depending continuously on the initial metric. A key constituent is the large-time diameter bound $\mathrm{diam}_{g_t}X<C\sqrt{t}$ (Bamler's Question 1.7). The expected payoff is a single Ricci-flow proof that $\mathrm{Isom}(X,g_X)\simeq\mathrm{Diff}(X)$ for every geometrizable prime 3-manifold, replacing the case-by-case topological proofs of Hatcher and McCullough–Soma.

Background

Let $X$ be a closed, connected, orientable 3-manifold, $\mathrm{Diff}(X)$ its group of smooth diffeomorphisms with the $C^\infty$-topology, and $\mathrm{Isom}(X,g)$ the isometry group of a Riemannian metric $g$. Thurston's geometrization theorem, proved by Perelman, states that every closed orientable 3-manifold decomposes along spheres and incompressible tori into pieces modeled on one of eight model geometries: $S^3$, $\mathbb{E}^3$ (flat), $S^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$, $\widetilde{SL(2,\mathbb{R})}$ (the universal cover of the unit-tangent-bundle group of the hyperbolic plane), $\mathrm{Nil}$ (the Heisenberg group with a left-invariant metric), $\mathrm{Solv}$ (a solvable non-nilpotent Lie group with left-invariant metric), and $\mathbb{H}^3$. A manifold is Haken if it is prime and contains an embedded, two-sided, incompressible surface; manifolds modeled on $\mathbb{H}^2\times\mathbb{R}$ or $\widetilde{SL(2,\mathbb{R})}$ are Seifert fibered spaces over a hyperbolic base orbifold (a compact base orbifold of negative orbifold Euler characteristic — with underlying surface of negative Euler characteristic, or the sphere with three cone points in the non-Haken case), and manifolds modeled on $\mathrm{Solv}$ are torus bundles over $S^1$ with Anosov monodromy or unions of two twisted $I$-bundles over the Klein bottle (all Haken). The generalized Smale conjecture asserts that if $g_X$ is a metric on $X$ locally isometric to its Thurston model geometry, then $\mathrm{Isom}(X,g_X)\hookrightarrow\mathrm{Diff}(X)$ is a homotopy equivalence.

The conjecture was settled case by case over four decades: Hatcher (1976) handled Haken manifolds; Gabai (2001) handled hyperbolic manifolds; Hong–Kalliongis–McCullough–Rubinstein (2012) handled elliptic (spherical) manifolds; McCullough–Soma (2013) handled Seifert fibered spaces with hyperbolic base orbifold, i.e. the $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL(2,\mathbb{R})}$ geometries including the non-Haken case. The remaining case — compact orientable non-Haken manifolds modeled on $\mathrm{Nil}$ — was proved by Bamler and Kleiner (2021, published 2023), which completed the determination of the homotopy type of $\mathrm{Diff}(X)$ for every closed, orientable, prime 3-manifold. Their spherical and hyperbolic cases also have Ricci-flow proofs (Bamler–Kleiner 2017 and 2019), so singular Ricci flow now yields a uniform approach covering spherical space forms, hyperbolic manifolds, $S^2\times S^1$, and non-Haken Nil manifolds.

A singular Ricci flow is the canonical Ricci flow with surgeries performed at infinitesimal scale, constructed for every closed 3-manifold initial condition and unique up to isometry (Kleiner–Lott 2017; Bamler–Kleiner 2022). The engine of the Nil theorem is the uniform asymptotic statement [Bamler–Kleiner 2023, Theorem 2.9(4) of arXiv:2108.03302]: for $X$ modeled on $\mathrm{Nil}$ and any initial metric $g_0$, for every $\epsilon>0$ there is a time $T(g_0,\epsilon)<\infty$, depending continuously on $g_0$ in the $C^2$-topology, such that the flow is non-singular on $[T,\infty)$ and every time-slice $(M_t,g_t)$ is $\epsilon$-almost-flat, meaning $\sup|Rm|\cdot\mathrm{diam}^2\le\epsilon$. Combined with a uniform almost-flat persistence theorem and a canonical improvement construction, the flow deformation-retracts the space of all metrics to the space of Nil-model metrics, which is then shown to be contractible; a fibration with fiber $\mathrm{Isom}(X,g_X)$ transfers this to the diffeomorphism group.

What is missing for the remaining geometries is exactly the long-time asymptotics. Bamler's long-time behavior series (2018), proved for Ricci flow with surgery, establishes finiteness of surgery times, the Type-III curvature bound $|Rm|<Ct^{-1}$ for large $t$, and dichotomies for $T^3$, Nil, and Solv manifolds: for a Solv manifold, the metrics $g_t$ are eventually well approximated by metrics with an isometric $T^2$-fiber structure, and if the normalized diameter $t^{-1/2}\mathrm{diam}_{g_t}$ stays bounded, then they are close to quotients of left-invariant Solv metrics. Whether that diameter bound always holds is open (Bamler's Question 1.7: if the geometric decomposition of $M_0$ is a single component, is $\mathrm{diam}_{g_t}M_0<C\sqrt{t}$ for all large $t$?). For the remaining Seifert geometries, the expected picture — sectional curvature pinched between $-(1/2+o(1))t^{-1}$ and $(1/4+o(1))t^{-1}$, the rescaled metrics $t^{-1}g_t$ collapsing along Seifert fibers of diameter $<\epsilon\sqrt{t}$ to the standard hyperbolic orbifold metric of curvature $-1/2$ on the base, and pointwise convergence of $t^{-1}g_t$ to a possibly singular limit metric (Bamler's Questions 1.8–1.10) — is likewise open. Lott (2007) proved that three-dimensional homogeneous solutions admitting finite-volume quotients have long-time limits given by expanding solitons (self-similar, homothetically expanding solutions of the Ricci flow equation on a homogeneous space), and, more generally, that with the diameter bound his compactness machinery yields an understanding of the subsequential Gromov–Hausdorff limits of $(X,t^{-1}g_t)$ and of their universal covers, so the diameter bound is the crux. Only the hyperbolic case is fully settled, where $(1/4)t^{-1}g_t$ converges to the hyperbolic metric; Kleiner's ICM 2022 survey states that this convergence-to-the-geometric-structure expectation 'has been confirmed only when M admits a hyperbolic metric'.

Problem Statement

Let $X$ be a closed, orientable, geometrizable 3-manifold, let $g_0$ be an arbitrary smooth Riemannian metric on $X$, and let $\mathcal{M}$ be the singular Ricci flow with initial time-slice $(X,g_0)$, which exists and is unique up to isometry. Establish the long-time asymptotics of $\mathcal{M}$ on the remaining non-flat Thurston geometries and use them to extend the Bamler–Kleiner Ricci-flow program to those geometries; the source additionally states that its methods can be carried further to all geometrizable 3-manifolds. Concretely, prove the analogues of Theorem 2.9(4) of the source paper (arXiv:2108.03302) in the following cases.

(1) $X$ modeled on $\mathbb{H}^2\times\mathbb{R}$ or $\widetilde{SL(2,\mathbb{R})}$ (a Seifert fibered space over a hyperbolic base orbifold, including the non-Haken case of a base sphere with three cone points): for every $\epsilon>0$ there is a time $T(g_0,\epsilon)<\infty$, depending continuously on $g_0$ in the $C^2$-topology, such that $\mathcal{M}$ is non-singular on $[T,\infty)$, the sectional curvatures of $g_t$ satisfy the model pinching $-(1/2+\epsilon)t^{-1}<\sec<(1/4+\epsilon)t^{-1}$ (the sectional pinching of Bamler's Question 1.9), and, on larger and larger neighborhoods as $t\to\infty$, the rescaled metrics $t^{-1}g_t$ collapse along the Seifert fibers, which have diameter $<\epsilon\sqrt{t}$, to the base orbifold, in the pointed Gromov–Hausdorff sense, the limit being the standard hyperbolic orbifold metric of curvature $-1/2$.

(2) $X$ a $\mathrm{Solv}$ manifold (a torus bundle over $S^1$ with Anosov monodromy, or a union of two twisted $I$-bundles over the Klein bottle): remove the bounded-normalized-diameter hypothesis in Bamler's Theorem 1.4(d) by proving the diameter bound $\mathrm{diam}_{g_t}X<C\sqrt{t}$ for all sufficiently large $t$ (Bamler's Question 1.7 in this case), and conclude that for every $\epsilon>0$ there is a time $T(g_0,\epsilon)$, depending continuously on $g_0$ in the $C^2$-topology, such that $\mathcal{M}$ is non-singular on $[T,\infty)$ and each time-slice is $\epsilon$-close, after rescaling by $t^{-1}$, to a quotient of a left-invariant metric on $\mathrm{Solv}$.

(3) Application to diffeomorphism groups: using these asymptotics to extend the Bamler–Kleiner Ricci-flow program — whose Nil case proceeds via canonical improvement of almost-model metrics to model metrics, contractibility of the space $\mathrm{Met}_{\mathcal{G}}(X)$ of model metrics, and the fibration $\mathrm{Diff}(X)\to\mathrm{Met}_{\mathcal{G}}(X)$ with fiber $\mathrm{Isom}(X,g_X)$ — prove that for every closed orientable prime 3-manifold $X$ with geometry $\mathcal{G}$ among $\mathbb{H}^2\times\mathbb{R}$, $\widetilde{SL(2,\mathbb{R})}$, and $\mathrm{Solv}$, the inclusion $\mathrm{Isom}(X,g_X)\hookrightarrow\mathrm{Diff}(X)$ is a homotopy equivalence for a model metric $g_X$, giving a uniform Ricci-flow proof in the cases currently known only by the topological methods of McCullough–Soma and Hatcher. The source's stated further goal is to carry the program to all geometrizable prime 3-manifolds. The diameter bound of Bamler's Question 1.7 for a general single-component non-hyperbolic geometric decomposition is an admissible, and expectedly necessary, constituent of a complete solution.

Scope and conventions: the task concerns the geometries $\mathbb{H}^2\times\mathbb{R}$, $\widetilde{SL(2,\mathbb{R})}$, and $\mathrm{Solv}$, addressed in items (1)–(3); the flat geometry $\mathbb{E}^3$ is not among them and falls, like the extension to arbitrary geometrizable 3-manifolds with mixed JSJ decompositions, under the source's stated further goal ('We believe that our methods can be even carried further to cover all geometrizable cases; this would be more technical, however'). The '$\epsilon$-close' and 'on larger and larger neighborhoods' convergence clauses in (1)–(2) are to be understood in the sense of the corresponding statements of Bamler's long-time behavior series: for the Solv conclusion, Theorem 1.4(d) there formulates closeness via an auxiliary metric $g'_t$ that is $(1+\epsilon)$-bilipschitz close to $g_t$ with $t^{-1}g'_t$ $\epsilon$-close to $t^{-1}g_t$ in the $C^{[\epsilon^{-1}]}$-sense, while for the Seifert conclusion the collapse in (1) is the pointed Gromov–Hausdorff convergence of $(X,t^{-1}g_t)$ to the base orbifold with the standard hyperbolic orbifold metric of curvature $-1/2$, the natural candidate indicated by Bamler's Questions 1.8–1.10.

References

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