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Hamilton's Ricci flow convergence theorems generally deal with metrics whose Ricci curvature is positive semidefinite. Here, we exhibit a nontrivial class of three-dimensional Riemannian metrics with Ricci curvature of indefinite sign for which the Ricci flow converges. Recent work of Hamilton [5], [6] shows that for all 3-dimensional Riemannian geometries (Σ, g) with positive Ricci curvature, the "Ricci flow" generated by the (heat-like) equation 0) Q-tgij = -2Rij+ 3 gijr (for r := fvRdμ/ fΣ3 dμ, the average of the scalar curvature R over Σ ) converges asymptotically in parameter time t to a metric of constant positive curvature. The nature of the proof of this result has led to speculation that for Riemannian geometries with Ricci) curvature of indefinite sign, the Ricci flow would generally not converge. The product geometry S x Sι9 whose Ricci flow approaches a singular "pinching" geometry, provides an example of the expected behavior. Here, we consider a class of 3-dimensional geometries, all with nondefinite or negative Ricci curvature, for which the Ricci flow does indeed always converge. This class, which we shall label ^ , is fairly specialized. All its members are topologically Γ 3 (3-torus) and all are invariant under a freely acting T isometry group. (There are a few other orthogonality conditions which characterize &\ we spell them out in §1.) The class is nontrivial, however, and as we shall prove in §2, all metrics in & converge (under Ricci flow) to a flat metric on Γ. The convergence proof, though fairly intricate in detail, has a single theme: showing that the scalar curvature R decays sufficiently rapidly as Received March 2, 1988. This work was partially supported by the National Science Foundation under Grant DMS8706494 at Oregon. 250 MAURO CARFORA, JAMES ISENBERG & MARTIN JACKSON / —• oo. We work with the scalar curvature because, in ^-metrics, it essentially controls all of the curvature. The proof (as detailed in §2) proceeds by first showing that R decays uniformly at the rate C\/(C2 + t) (here and elsewhere, Cinteger denotes some positive constant). This guarantees that the metric goes flat, but we need faster decay to prove convergence. To get this more rapid decay, we establish that the areas of the T-orbits (of the isometry group) converge as t —> oo and thence the lengths of the orthogonal S paths converge as well. This allows us to do the remaining analysis on S, with a finite (converging) total perimeter length for all time. Using various estimates on the circle to simplify the evolution equation for i?, we are able to show that the volume average of R decays exponentially and then with a few more estimates we show that R uniformly decays to zero exponentially. Smooth convergence of the metric then follows. An interesting feature of the Ricci flow of the & metrics is that for them—unlike the Ricci-positive metrics studied by Hamilton—the limit metric is not easily predicted. The Ricci flow generated by (1), we recall, is "normalized" so that it preserves total volume of (Σ, g) along the flow. Since for a given compact manifold Σ there is generally at most one constant positive curvature metric of a given fixed volume, the limit of the Ricci flow of a Ricci-positive metric on such a manifold is fixed. There are, however, many flat metrics of a given volume on Γ, and the Ricci flow of a & metric may converge to any one of them. One possible consequence of this (continuous) family of possible Ricci flow limits is that for some metrics on Γ 3 (or on other topologies which allow flat metrics) the Ricci flow may not converge to a single flat metric but instead may "quasiconverge" to a one-parameter family of them. Bits of our proof, and some results of Gray son and Hamilton [4], support this possibility. We comment on it and other issues related to attempts to extend our results in §3. While our results should be useful in furthering the general understanding of Ricci flow, the motivation for our work comes from general relativity. We are interested in setting up a program for approximating the evolution of cosmological spacetime solutions of Einstein's equations via the development of a procedure for "smoothing" sets of initial data for such spacetimes [1]. The Ricci flow could be an important part of this smoothing. As a testing laboratory for this study, we are examining the fairly well-understood "Gowdy" [3] spacetime models. The initial data for the "polarized" Gowdy models involve metrics of the class &. Hence it is important to our program that the Ricci flow for & metrics converges. CONVERGENCE OF THE RICCI FLOW 251 1. The ^-metrics and their Ricci flow equations Here, we define the class of metrics ( " ^ metrics") with which we shall be working, set up coordinates in which to study them, calculate their curvatures, and write out the equations which govern their Ricci flow. Definition (^-metrics). For each (Σ, g) € «y, we require the following. (a) Σ = T\ (b) g is invariant under a free T action on Γ. (c) The killing vector fields generating the T action may be chosen to be (globally) orthogonal and independent. (d) The areas of all the T orbits are the same. (e) Each line in the one-dimensional congruence of paths orthogonal to the T orbits is closed (a circle). All of these properties together allow us to pick periodic coordinates (x,y, θ) so that dx and dy are the killing fields generating the T 2 isometry, and the metric takes the form ~ (2) g = edθ + ef[edx + e where a is a function of θ, W is a function of θ, and / is a constant (all three will also be functions of t when we consider the Ricci flow of g). Since θ is the only noncyclic coordinate, all of our analysis is done on the circle parametrized by θ. Hence it is useful to define the arc-length parameter s on this circle, (3) 5(0)= / edθ, Jo with the corresponding arc-length one-form (4) ds = edθ and arc length vector field (5) ds = e~ adθ. In terms of these, we have simple expressions for the vector calculus quantities on Σ; e.g. for any scalar Ψ on Σ we find (6) VΨ VΨ=(<95Ψ) 2 and VΨ = <95 Ψ. Note also that, so long as we choose x e [0,1) and y e [0,1), the volume 3-form dμ on Σ 3 may be effectively treated as a 1-form (7) dμ = edθ use the notation dx = d/dx etc., here and throughout the paper. 252 MAURO CARFORA, JAMES ISENBERG & MARTIN JACKSON with integration over x and y understood. For convenience in doing Fourier series below, we shall let the range of θ be 0 to 2π. From (2), we readily calculate the curvature of the ^ metrics. Since the Ricci curvature of a 3-dimensional Riemannian manifold determines all of its curvature, we list only that (in coordinate basis): (8a) Ree = -{e2a(dsW) ,

Convergence of the Ricci flow for metrics with indefinite Ricci curvature