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Ricci flow of non-collapsed 3-manifolds whose Ricci curvature is bounded from below

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme. Given such initial data (M,g) we show that a Ricci flow exists for a short time interval. This enables us to construct a Ricci flow of any (possibly singular) metric space (X,d) which arises as a Gromov-Hausdorff limit of a sequence of 3-manifolds which satisfy a), b) and c) uniformly. As a corollary we show that such an X must be a manifold.


Introduction and statement of results
A smooth family of metrics (M, g(t)) t∈[0,T ) is a solution to the Ricci flow if ∂ ∂t g(t) = −2Ricci(g(t)) ∀ t ∈ [0, T ).
More specifically, we consider the class of smooth, complete Riemannian manifolds (M, g) which satisfy (a) Ricci(g) ≥ k (b) vol( g B 1 (x)) ≥ v 0 > 0 for all x ∈ M It is well known, see [23], that every sequence of smooth Riemannian manifolds satisfying (a) contains a subsequence which converges with respect to the Gromov-Hausdorff distance to a possibly singular metric space (M, d) (see [4] for a definition of Gromov-Hausdorff distance: this distance is a weak measure of how close metric spaces are to being isometric). With the expression 'possibly singular' we mean two things: • it is possible that the limiting space (M, d) is no longer a manifold (see Example 1.1 below) and • it is possible that the resulting metric d is not smooth, even if M is a manifold (see Example 1.2 below). with the quotient topology, where l((r, x), (s, y)) := r 2 + s 2 − 2rs cos(γ(x, y)) for all r, s ∈ R + 0 and all x, y ∈ RP 3 and γ : RP 3 × RP 3 → R + 0 is the standard distance on RP 3 . In particular, N is not a manifold.
This is because: in dimension two Ricci ≥ −k 2 =⇒ sec ≥ −2k 2 . Then a Theorem of G.Perelman says that (M, d) is a manifold: see [32].
So we see that in dimension two any metric space (M, d) which arises as the GH limit of a sequence of Riemannian manifolds satisfying In this paper we obtain as a consequence of one of our main theorems (Theorem 9.2 in this paper) that this conjecture is correct, if each of the manifolds occuring in the sequence is compact or we demand that the geometry at infinity is controlled in a certain sense (see condition (c) andc below). That is we will assume that each of the manifolds (M, g) occuring in the sequence satisfy additionally (c) sup M |Riem(g)| < ∞ (bounded curvature) or (c ) Let f : M → R be the exponential function composed with its self mtimes, and ρ : M → R + 0 the distance function from a fixed base point b, ρ(x) := dist(x, b). We assume that (c 1 ) ρ : M − B R (b) → R is smooth for some R > 0, and k − concave there, that is |Riem(x)|/f (r)) = 0.
Remark 1.5. Note that condition (c) is trivially satisfied if M is compact.
Remark 1.6. Assume (c 2 ) is satisfied for some m ∈ N , and that the sectional curvatures of (M, g) are larger than −l on M and that cut (b) ∩ (M − B R (b)) = ∅ for some R > 0. Then condition (c 1 ) is satisfied for some k = k(n, l) and some larger m (depending on the initial m), as as one sees using the hessian comparison principle (see for example chapter 1 of [44]).
Under these restrictions, we obtain that the conjecture of Anderson-Cheeger-Colding-Tian is correct. That is, we prove: be a metric space arising as the GH limit of a seqeunce of three dimensional Riemannian manifolds (M i , g i ), i ∈ N each of which satisfies (a), (b) and (c) or each of which satisfies (a), (b) and (c). Then X is a three dimensional manifold. If furthermore each of the (M i , g i ) has diameter bounded above by a uniform constant d 0 < ∞, then M i is diffeomorphic to X for all i sufficiently large.
Remark 1.8. In the case that all manifolds in the sequence above satisfy a two sided Ricci curvature bound, |Ricci| ≤ k 2 , a bound on the integral of the curvature tensor M |Riem| 3/2 ≤ D and (b) is satisfied, M.Anderson also proved that the limit space X is a manifold: see Corollary 2.8 in [2]. Later, Cheeger-Colding-Tian (see [8] Theorem 1.15) proved that the singular set of the limit space (X, d X ) is empty, if all manifolds occuring in the sequence above satisfy (a),(b) and − B1(x) |Riem| 3/2 ≤ D for all balls of radius one. The condition M |Riem| 3/2 ≤ D prohibits non-flat cones over spheres occuring in (X, d). Theorem 1.7 allows the occurence of such cones.
The method we use to prove this theorem is as follows. Let (M i , g i (0)) be a sequence of manifolds satisfying (a),(b) and (c). We flow each of the (M i , g i (0)) by Ricci flow to obtain solutions (M i , g i (t)) t∈[0,Ti) . Then we prove uniform estimates (independent of i) for the solutions. Once we have these estimates, we are able to take a limit of these solutions, to obtain a new solution (M, g(t)) t∈(0,T ) where M is some manifold. This solution will also (by construction: it is a smooth limit) satisfy similar estimates to those obtained for (M i , g i (t)) t∈[0,Ti) . Using these estimates, we show that (M, d(g(t))) → (X, d X ) in the Gromov-Hausdorff sense as t ց 0, and that in fact X is diffeomorphic to M . The most important step in this procedure is proving uniform estimates for the solutions (M i , g i (t)) t∈[0,Ti) . The case that the (M, g i (0)) satisfy (a),(b) and (c) is reduced to the case that the (M, g i (0)) satisfy (a),(b) and (c) by a conformal deformation of the starting metrics (which leave the starting metrics unchanged on larger and larger balls as i → ∞: see chapter 7 for details).
The estimates we require to carry out this procedure are obtained in the following theorem (see Theorem 9.1). Theorem 1.9. Let k ∈ R, 0 < v 0 ∈ R, m ∈ N and (M, g 0 ) be a three (two) manifold satisfying (a),(b) and (c) with constants k, v 0 and m respectively. Then there exists a T = T (v 0 , k, m) > 0 and K = K(v 0 , k, m) > 0 and a solution (M, g(t)) t∈[0,T ) to Ricci-flow satisfying To help us prove Theorem 1.9 we prove estimates on the rate at which the infimum of the Ricci curvature can decrease, and on the rate at which the distance function and volume of such a solution can change (see Lemma 6.1 and 6.2). As an application of Theorem 1.9 and these estimates we get (Theorem 9.2 in this paper). Theorem 1.11. Let k, v 0 , m ∈ R be fixed. Let (M i , i g 0 ) be a sequence of three (or two) manifolds satisfying (a),(b),(c) or (a),(b),(c), (with constants k, v 0 , m indepedent of i) and let (X, d, x) = lim i→∞ (M i , d( i g 0 ), x i ) be a pointed Gromov-Hausdorff limit of this sequence. Let (M i , i g(t)) t∈[0,T ) be the solutions to Ricci-flow coming from the theorem above. Then (after taking a sub-sequence if necessary) there exists a Hamilton limit solution (M, g(t), x) t∈(0, In particular, X is a manifold. As a corollary to this result and Lemma 5.1 we obtain the following corollary (Corollary 9.4 in this paper).
be a sequence of three (or two) manifolds satisfying (b), (c) or (b),(c), and Let (X, d X ) = GH lim i→∞ (M i , d( i g 0 )) (notation GH lim refers to the Gromov Hausdorff limit). Then the solution (M, g(t), x) t∈(0,T ) obtained in Theorem 1.11 satisfies Ricci(g(t)) ≥ 0 for all t ∈ (0, T ) and (X, d X ) is diffeomorphic to (M, g(t)) for all t ∈ (0, T ). In particular, combining this with the results of W.X.Shi [46] and R.Hamilton [24], we get that (X, d X ) is diffeomorphic to R 3 , S 2 × R or S 3 modulo a group of fixed point free isometries in the standard metric.

Previous results
We present here some previous results related to Ricci flow of non-smooth metrics.
In the paper [47], the Ricci flow of continuous metrics is considered. Estimates similar to those in Theorem 1.9 are proved.
In the paper [13] Kähler Ricci flow of L ∞ Kähler metrics is considered, estimates similar to (a) t of Theorem 1.9 are proved.
In the paper [54] the Author considers the Ricci flow of initial metrics which satisfy ( uniformly) an L n/2 bounds on the curvature, an L p bound on the Ricci curvature (p > (n/2)) and a volume and diameter bound. He proves using Moser iteration, that estimates similar to (a) t of Theorem 1.9 hold under the Ricci flow of such a metric.
In the paper [11], the class of metrics with |Ricci| ≤ 1 and conjugate radius bigger than r 0 is considered. The authors prove estimates similar to (a) t of Theorem 1.9 once again using moser iteration.
In the paper [39], the author proves an estimate of the form (a) t of Theorem 1.9, under the assumtion that all neighbourhoods are alomost euclidean, and the scalar curvature is bounded from below. Here, a blow up argument is used, and an analysis of a backward evolving heat-type flow (see also [41] and [5] ).
In the paper [21], the author extends the results of Yang to the case that the manfiold is non-compact, and Ricci ≥ −1 and an L p bound on the curvature holds (p > (n/2)) (see also [34]).
The case that the L (n/2) curvature is small locally, and and an L p bound on the norm of the Ricci curvature exists, is considered in the paper [53].

Methods and structure of this paper
As explained in the introduction, we shall cheifly be concerned with Riemannian manifolds (M, g) which are contained in T (3, k, m, v 0 ) or T ∞ (3, k, v 0 ), where these two spaces are defined as follows.
We say (M, g) ∈ T (n, k, m, v 0 ) if (a) and (b) are satisfied and the condtion (c) is replaced by (c) Let f : M → R be the exponential function composed with its self mtimes, and ρ : M → R + 0 the distance function from a fixed base point b, ρ(x) := dist(x, b). We assume that |Riem(x)|/f (r)) = 0.
Let us defineT (n, k, m, v 0 ) (T ∞ (n, k, v 0 ) ) as the set of metric spaces (X, d X ) which arise as the Gromov-Hausdorff limit of sequences whose elements are contained in T (n, k, m, v 0 ) ( T ∞ (n, k, v 0 ) ). Elements ofT (n, k, m, v 0 ) (T ∞ (n, k, v 0 ) ) can be very irregular, and are not a priori manifolds (as we saw in the two examples of the introduction). Nevertheless, they will be length spaces and do carry some structure. In the first part of the paper we concern ourselves only withT ∞ (3, k, v 0 ).
. In order to define a Ricci flow of (X, d X ) we will flow each of the (M 3 i , g i ) and then take a Hamilton limit of the solutions (see [29]). The two main obstacles to this procedure are: • It is possible that the solutions (M i , g i (t)) are defined only for t ∈ [0, T i ) where T i → 0 as i → ∞. • In order to take this limit, we require that each of the solutions satisfy uniform bounds of the form for some well defined common time interval (0, T ) and some function c : is allowed). Furthermore they should all satisfy a uniform lower bound on the injectivity radius of the form inj(M, g i (t 0 )) ≥ σ 0 > 0 for some t 0 ∈ (0, T ). As a first step to solving these two problems, in Lemma 4.3 of Section 4 we see that a (three dimensional) smooth solution to the Ricci flow (M, g(t)) t∈[0,T ) such that (M, g(t)) ∈ T ∞ (3, k, v 0 ) for all t ∈ [0, T ) and sup M×[0,S] |Riem| < ∞ for all S < T cannot become singular at time T . Furthermore, a bound of the form for such solutions is proved: that is, the curvature of such solutions is quickly smoothed out.
In Section 5 we prove an a priori estimate on the rate (Lemma 5.1) at which the infimum of the Ricci curvature of a solution to the Ricci flow with bounded curvature can decrease. Note: this lemma is a non-compact version of Lemma 5.1 from [48].
One of the major applications of this Lemma is: any solution (M, g(·, t)) t∈[0,T ) in T ∞ (3, k, v 0 ) which has bounded curvature at all times and satisfies Ricci(g 0 ) ≥ −ε 0 at time zero, must also satisfy R(g(t)) ≤ c0 t (from (3.1)) and hence from (3.2) In Section 6, we consider smooth solutions to the Ricci flow which satisfy In Lemma 6.1, well known bounds on the evolving distance for a solution to the Ricci flow are proved for such solutions.
We combine this Lemma with some results on Gromov-Hausdorff convergence and a theorem of Cheeger-Colding (from the paper [10]) to show (Corollary 6.2) that such solutions can only lose volume at a controlled rate.
The results of the previous sections are then used to prove a theorem ( section 6 ) which tells us how a priori the Ricci flow of an element (M, g 0 ) ∈ T ∞ (3, k, v 0 ) behaves: see Theorem 7.1.
In Section 8 we show that any (M, g) ∈ T (n, k, m, v 0 ) can be approximated in the GH sense by manifolds (M, g i ) ∈ T ∞ (n, k,ṽ 0 ), i ∈ N. More precisely, we show that there existsṽ 0 =ṽ 0 (n, k, m, v 0 ) > 0 and (M i , g i ) ∈ T ∞ (n, k,ṽ 0 ) with such that (M, d(g i )) → (M, d(g)) in the Gromov-Hausdorff sense as i → ∞. This section is independent of the rest of the paper, and requires no knowledge of the Ricci flow.
Finally, using the results of the previous two sections, we show that a solution to the Ricci flow of (X, d X ) exists, where (X, d X ) is the Gromov-Hausdorff limit as , and that this solution satisfies certain a priori estimates. See Theorem 9.2.
Appendix A contains some Hessian comparison principles and the proofs thereof. Appendix B contains a result on the rate at which distance changes under Ricci flow if the solution satisfies |Riem| ≤ c/t.

4.
Bounding the blow up time from below using bounds on the geometry.
An important property of the Ricci flow is that: if certain geometrical quantities are controlled (bounded) on a half open finite time interval [0, T ), then the solution does not become singular as t ր T and may be extended to a solution defined on the time interval [0, T + ε) for some ε > 0. As in the paper [48], we are interested in the question: Problem 4.1. What elements of the geometry need to be controlled, in order to guarantee that a solution does not become singular?
In [45], it was shown that for (M, g 0 ) a smooth non-compact Riemannian manifold with sup M |Riem(g 0 )| < ∞, the Ricci flow equation (the compact case was proved by Hamilton in [24]). Using Shi's solution (Theorem It was also shown in Shi [45] that if (M, g(t)) t∈[0,T ) is a smooth solution with T < ∞ and sup M×[0,T ) |Riem| < ∞, then there exists an ε > 0 and a solution So we see that a bound on the supremum of the Riemannian curvature on M × [0, T ) (that is, control of this geometrical quantity) guarantees that this solution does not become singular as t ր T , and that it may be extended past time T (where we are assuming here that T < ∞). In the following lemma, we present other bounds on geometrical quantities which guarantee that a solution to the Ricci flow does not become singular as t ր T (once again, T < ∞ is being assumed here).
for all t ∈ [0, T ) ( notation: R refers to the curvature operator). Then there exists Proof. Assume to the contrary that there exist solutions (M i , i g(t)) t∈[0,Ti) , T i ≤ 1 to Ricci flow satisfying the conditions (i),(ii),(iii) and such that in this case we redefine M i = M j and T i = T j for all i ∈ N and hence we do not need to treat this case separately ) such that in view of the fact that Then for all sucht we have in view of the definition of (p i , t i ), and 0 ≤ s ≤ t i (follows from the definition of s and the fact thatt ≤ 0), and (4.3). Since vol(B 1 (p), i g(t)) ≥ v 0 > 0 and Ricci ≥ −k 2 (R ≥ −n 2 k 2 ) (in the case n = 3 this is true by assumption, in the general case it is true as all sectional curvatures are not less then −k 2 ) we have (in view of the Bishop Gromov comparison principle) which implies the same result (for radii scaled appropriately) for the rescaling of the manifolds: we obtain a bound on the injectivity radius from below, in view of the theorem of Cheeger/Gromov/Taylor, [12] the theorem of Cheeger/Gromov/Taylor says that for a complete Riemannian manifold (M, g) with |Riem| ≤ 1, we have for all r ≤ π 4 : in particular, using that diam (M, g) ≥ N ,N as large as we like, and |Riem| ≤ 2 for the Riemannian manifolds in question, we obtain inj(x, g) ≥ṽ 0 s n+1 ls n + ω n exp n−1 ≥ c 2 (ṽ 0 , n) > 0 for s = min((ω n exp n−1 ) 1 n , π 4 ) . This allows us to take a pointed Hamilton limit (see [29]), which leads to a Ricci flow solution (Ω, o, g(t) t∈(−∞,ω) ), with |Riem(g(t))| ≤ |Riem(o, 0)| = 1, and Ricci ≥ 0 (R ≥ 0), ω ≥ 0 In fact the limit solution satisfies R ≥ 0 for n = 3 also, see Corollary 9.8 in [15]. The volume ratio estimates are also valid for (Ω, g), in view of (4.5).
We now apply Proposition 11.4 of [39] to obtain a contradiction.

Bounds on the Ricci curvature from below under Ricci flow in three dimensions
The results of this section are only valid in dimensions two and three. We prove a quantitative estimate that tells us how quickly the Ricci curvature can decrease, if we assume at time zero that the Ricci curvature is not less than −1 and that the supremum of the curvature of the evolving metric is less than infinity. This involves modifying the argument from [48] to the case that M is noncompact. This result has similarities to the estimate of Hamilton-Ivey (see [28] or [31] for a proof of the Hamilton-Ivey estimate, which was independently obtained by R.Hamilton and T.Ivey). For a general heat type equation on a non-compact it is well known that the maximum principle does not hold for general solutions f , and for general V and a. In the case that a and V are bounded, there are a number of maximum principles which can be applied as long as the growth of f is controlled, and the evolving metric g satisfies certain conditions (for example | ∂ ∂t g| ≤ c ): see for example [19], [38]. In the case of Tensors, there are also a number of Theorems which present conditions which guarantee that the Tensor Maximum principle of Hamilton holds in a non-compact setting: see for example Theorem 2.1 in [37] and Theorem 7.1 of [47].
In the proof of the lemma below we construct a tensor L which satisfies ∂ ∂t L ≥ ∆L + N where L(·, 0) ≥ 0 and L(x, t) ≥ ε > 0 for all x far away from an origin, This allows us to argue exactly as in the proof of the Tensor maximum principle for compact manifolds (proved by R.Hamilton in [24]) to conclude that L ≥ 0 everywhere if L ≥ 0 at t = 0.
Define ε = ε(t) = ε 0 (1 + kt), and the tensor L(t) by )(x 0 , x) for some fixed x 0 , and a = 1000n(1 + sup M×[0,T ] |Riem(g(t))|). We will often write ε for ε(t) (not to be confused with ε 0 ). Notice that ε 0 < ε(t) ≤ 2ε 0 , for all t ∈ [0, 1 k ) = [0, 1 100 ) : we will use this freely. Then L i j = (R i j + εRtδ i j + εδ i j + σf δ i j ), and as in the paper [48], we calculate: and N ij is (up to the constant k = 100) the same as the Tensor from the paper [48], for ta ≤ 1 in view of the Laplacian comparison principle (see the Hessian comparison principle in Appendix A), as long as ρ 2 is smooth in time and space where we differentiate.
In the following, we argue as in the proof of Hamilton's maximum principle, Theorem 9.1, [24]. We claim that L ij (g(t)) > 0 for all t ∈ [0, T ). Notice that f has exponential growth, and the other terms in the definition of L are bounded. This guarantees that L > 0 outside a compact set. Hence if L ij (g(t)) > 0 is not the case, then there exist a first time and point (p 0 , t 0 ) and a direction w p0 for which L(g(t))(w p0 , w p0 )(p 0 , t 0 ) = 0.
Hence we may argue as above to obtain a contradiction. Now letting σ go to zero, we get Ricci ij + εRtg ij + εg ij ≥ 0 as long as ta ≤ 1 and tk ≤ 1. But then, we may argue as above starting at t 0 = 1 a , but now with f 1 in place of f , f 1 = e ρ 2 (1+a(t−t0))+a(t−t0) to obtain the same result on [0, 2t 0 ] as long as tk ≤ 1. Continuing in this way, we see that Ricci ij + εRtg ij + εg ij ≥ 0 as long as tk ≤ 1.
The case for the sectional curvatures is similar: from [25], Sec. 5, we know that the reaction equations for the curvature operator are It is shown in [48] (in the proof of the compact version of this Lemma) that (for ε(t) : at the points where f is smooth and ta ≤ 1. So the ordinary differential equation for f satisfies at the points where f is smooth and ta ≤ 1.
Since f is exponential in distance, the points where α + εtR + ε + σf ≤ 0 is a compact set. Hence, if α + εtR + ε + σf > 0 is not true, then there must exist a first time and point (p 0 , t) where this fails. At such a point (p 0 , t) we have (from (5.3) and (5.4)): as long as ta ≤ 1, where we have used that α + εtR + ε = −σf , and that |γ| ≤ a 100 . Using α + εtR + ε = −σf again, we get since R ≥ −3ε 0 is preserved by the flow, and t ≤ 1 k . Hence, inserting this into (5.5) we get ∂ ∂t (α + εtR + ε + σf ) > 0 at a point where α + εtR + ε + σf = 0. Choose an orthonormal basis for the two forms at (p 0 , t 0 ): (time independent by definition) for which the curvature operator is diagonal, and assume that R(φ 1 , φ 1 ) = R ijkl φ 1 ij φ 1 ij is the smallest Eigen-value of the curvature operator R. Then we have ∂ ∂t Using the maximum principle, we obtain the result by arguing as in the case of the Ricci curvature above (once again, if this inequality is violated at some point and first time, then we may need to modify ρ in order to make sure that it is smooth, as in the argument above for the Ricci curvature).

Bounding the distance and volume growth in terms of the curvature
The results of this section hold for all dimensions. Ricci(g(t)) ≥ −1, |Riem(g(t))|t ≤ c 0 . Proof. These results essentially follow from [28], theorem 17.2 (with a slight modification of the proof suggested by the editors in [6] : see Appendix B) and [28], Lemma 17.3: see Appendix B for a proof. Corollary 6.2. Let (M n , g(t)) t∈[0,T ) be an arbitrary smooth solution to Ricci flow (g(0) = g 0 ) satisfying the conditions 6.1 and assume that there exists v 0 > 0 such that Then there exists an S = S(c 0 , v 0 , n) > 0 such that Notice that this then implies in view of the Bishop Gromov comparison principle.
Proof. If this were not the case, then there exist solutions (M n i , i g(t)) t∈[0,Ti) satisfying the stated conditions and there exist t i ∈ [0, T i ), t i i→∞ −→ 0 and points p i ∈ M i such that vol(B 1 (p i , t i )) < 2v0 3 . A subsequence of (M i , d( i g(0)), p i ) converges to (Y, d, p) in the pointed Gromov-Hausdorff limit. Clearly then (M i , d( i g(t i )), p i ) also converges to (Y, d, p), in view of the characterisation of Gromov Hausdorff convergence given in Corollary 7.3.28 of [4], and the estimates (6.2) (since t i → 0). The Theorem of Cheeeger and Colding says that volume is continuous under the limit of non-collapsing spaces with Ricci curvature bounded from below: But this is a contradiction as we then have 2v 7. Non collapsed non compact three manifolds with curvature bounded from below.
The results of this section are only valid for dimensions two and three-Theorem 7.1. Let (M, g 0 ) be a complete smooth three (or two) manifold without boundary in T ∞ (3, k, v 0 ): that is Then there exists an S = S(v 0 , k) > 0 and K = K(v 0 , k) and a solution (M, g(t)) t∈[0,T ) to Ricci-flow which satisfies T ≥ S, and (note that the estimates are trivial for t = 0).
Proof. We assume n = 3. The argument for n = 2 is the same. Before proving the theorem rigorously, we present a sketch of the proof which leaves out the technical details. This should give the reader a clear picture of the structure of the proof. As a first step, we scale the metric by a large constant, so that Ricci(g 0 ) ≥ −ε for a small ε = ε(v 0 , k) > 0. The condition vol( B r (x)) ≥ṽ 0 r 3 ∀ 0 < r ≤ 1, (7.5) for someṽ 0 =ṽ 0 (v 0 , k) > 0. Rescale the metric by the constant 1000c 0 so that Ricci(g 0 ) ≥ −ε where ε = 1 1000c0 and c 0 = c 0 (3, −1,ṽ 0 2 ) is from the Lemma 4.3. Notice that (7.5) is still true for this new rescaled metric, as we have scaled by a constant which is larger than 1. We denote our rescaled metric also by g 0 .
From the work of W.Shi (see [45], main Theorem) we know that there exists a solution (M, g(t)) t∈[0,T ) to Ricci flow, with g(0) = g 0 , sup M |Riem(g(t))| < ∞ for all t ∈ [0, T ). Without loss of generality, (M, g(t)) t∈[0,T ) is a maximal solution with BC in the sense of Definition 4.2. Let T M be the supremum over all S ≤ T such that: for all t ∈ [0, S) ∩ [0, T ). First we show that T M > 0. We have bounded curvature on compact time intervals, N (δ) := sup M×[0,T −δ] |Riem(g(t))| < ∞, and hence for t ≤ K, K = K(N, ε) small enough (see argument in the proof of Lemma 5.1) at all points where ρ 2 is differentiable. Since Ricci + ε exp (ρ 2 (1+at)+at) g must take its infimum at an interior point, we get (arguing as in the proof of Lemma 5. Ricci ≥ −1 for all t ≤ S. Now we rescale the metric back, to obtain the result: the rescaled solution h(·,t) = (1/1000c 0 )g(·, 1000c 0t ) is the desired solution. Its' initial value is given by g 0 (g 0 is as in the beginning of the proof of the theorem) and it satisfies the required estimates by scaling ( |Riem ≤ c0 t is a scale invariant inequality, and the estimate 'Ricci ≥ −1 for all t ≤ S' scales to 'Ricci ≥ −1000c 0 for all t ≤ S/(c 0 1000)'). That the volume of a ball of radius one is larger than v 0 /2 for the evolving metric follows from the corollary of the previous section (after shortening the time interval if necessary). The estimate (d t ) follows immediately from Lemma 6.1.

Conformal deformation of non-collapsed manifolds with
Ricci ≥ −1 Let (M, g) be a manifold satisfying (b) and (c): We wish to modify the metric g to a new metric g i so that • where b is a fixed origin and i ∈ N. In the next section we will apply the results of the previous sections in order to flow the g i 's, and then we will take a limit in i of the resulting solutions.
For convenience, we introduce the following notation. , where the function on the right hand side is the composition of exp m times, and m is a fixed number in N. We call functions of the type appearing in the right hand side an exponential comparison function.
We require the following help Lemma about exponential comparison functions. Lemma 8.2. Let h be an exponential comparison function, 0 < k ∈ R, 0 ≤ p ∈ R.
We have |r| p + |h(r)| p + |h ′ (r)| p + |h ′′ (r)| p ≤ c k,p e kh(r) for all r ∈ R + , |h i (y)| ≤ c k e khi(y) for all y ∈ R + , |h ′ i (y)| ≤ c k e khi(y) for all y ∈ R + , |h ′′ i (y)| ≤ c k e khi(y) for all i < y ∈ R + , for some constants c p , c k,p , depending on p, respectively k and p, and the function h but not on i.
Proof. The first estimate follows from the definition of an exponential comparison function h and the fact that |y| q ≤ c q,k e k|y| for q ≥ 0, for some constant c q,k . The next estimate follows from the definition of h i and the first estimate: The third estimate may be seen as follows where we have freely used the first estimate. Replacing k by (k/2) we obtain the desired estimate. The method for estimating (h i ) ′′ is similar: Theorem 8.4. Let (M, g) be a smooth Riemannian manifold with controlled geometry at infinity satisfying Then there exists a family of smooth Riemannian metrics {g i }, i ∈ N on M satisfying Proof. Let δ << 1 be fixed for the rest of this section. Let φ = h i for some i ∈ N, where h i is as in Lemma 8.2. Note that φ is a non-decreasing function φ : R + → R + . Let b be a fixed base point in M . ρ : M → R + is the distance function with respect to b: ρ(x) := dist(x, b). Let x 0 be an arbitrary point in M and set ρ 0 := ρ(x 0 ). Let γ : [0, ρ 0 ] → M be a minimizing geodesic from b to x 0 (with unit speed). So d(γ(s), γ(u)) = |u − s| for all u, s ∈ [0, ρ 0 ]. As M is complete, we may extend this smoothly to a geodesic γ : [0, ∞) → M. Let r > 0 be some positive number: later we will choose r to depend on ρ 0 , but at first we simply require r to be some positive radius. Let y 0 := γ(ρ 0 − (r/2)). In particular d(x 0 , y 0 ) = d(γ(ρ 0 ), γ(ρ 0 − (r/2))) = r/2. Due to the triangle inequality we have: In particular, using this inclusion and the fact that φ is non-decreasing, we have Now set r := e −(1/2)φ(ρ0) . We obtain a lower bound for φ(ρ(x)) − φ(ρ 0 ) for x ∈ B δr (y 0 ) as follows.
Defineg(x) := e φ(ρ(x)) g(x). Balls with respect tog will be denoted with a tilde: B s (p) is the ball with radius s and centre p ∈ M with respect tog. We denote distance with respect tog also with a tilde:d(x, y) is the distance with respect tõ g from x to y. The volume form with respect to g is denoted by dµ g , and that ofg with dµg. We wish to show that (M,g) is also non-collapsed. Let x ∈ M be given. Now set r = e −(1/2)φ(ρ0) .
The well known formulas for the change of the metric g tog = e f g = ψg (for example see equation 13 in Chapter 8 of [45]) for a function f : M → R (where here ψ is defined to be ψ(x) := e f (x)) arẽ where ∇l denotes the gradient of the function l, and (∇ 2 l) denotes the second covariant derivative of f (which is a ( 0 2 ) Tensor), both w.r.t to g. In the following g | · | will denote the norm with respect to g. Now let f be f (x) = φ(ρ(x)) (this implies ψ(x) = e f (x) = e φ(ρ(x)) ) where ρ is the distance function with respect to g, and φ : R → R is an arbitrary smooth function. Our assumption of controlled geometry at infinity implies that g |∇ρ| = 1 on M − B R (b) and one version of the Hessian comparison principle tells us that wherever ρ is differentiable and larger than one (see Appendix A ) and here R B : The following identities then follow from the definitions of f (x) = φ(ρ(x)), and ψ = e φ(ρ) = e f : for some universal constant c not depending on k and n (later we will examine different φ's but they all satisfy an estimate of the form above for the same constant c). Using |∇ρ| 2 = 1, (8.8), (8.9) and that ρ is k-concave, we get since g ij = e −fg ij and f > 0. We will assume in the following that lim r→∞ e −(1/8)φ(r) R B (r + 2) = 0, (8.11) where R B (r) is the function introduced above in 8.7. We estimate the equalities (8.8) using the growth properties of φ ((8.9)) as follows g |∇ψ(x))| 2 ≤ cψ 9/4 , g |∇ 2 ψ|(x) ≤ ψ 5/4 (x)c(ρ(x) + 2)(R B (ρ(x) + 2) + c(n)(k + 1)). (8.12) Hence, using formula (8.6), we get 13) in view of (8.11) and the fact that ψ(x) = e φ(ρ(x)) . Note that φ i satisfies is a metric satisfying as required.

Applications
Let (M, g 0 ) ∈ T (3, k, m, v 0 ) and let (M, i g 0 ) ∈ T ∞ (3, k,ṽ 0 ) be the smooth metrics constructed in the previous section : remember that these i g 0 satisfy i g 0 = g 0 for all x ∈ B i (p 0 ). Now we may apply Theorem 7.1 to each (M, i g 0 ) to obtain solutions (M, g i (t)) t∈[0,T (n,ṽ0)) ∈ T ∞ (3,k,ṽ 0 ) satisfying the a priori estimates. Hence using the local estimates of Theorem 1.3 of [49] and the interior estimates of Shi (see [6]), we may take a Hamilton limit to get a solution to Ricci flow (M, g(t) t∈[0,T ) ) which satisfies the a priori estimates (7.1). Note that the local estimates of Theorem 1.3 in [49] guarantee that we may take the limit on the interval [0, T ) and not just (0, T ). So we have proved: Theorem 9.1. Let (M, g 0 ) be a three (or two) manifold in T (3, k, m, v 0 ) (T (2, k, m, v 0 )). Then there exists a T = T (v 0 , k, m) > 0 and a solution (M, g(t)) t∈[0,T ) to Ricci flow, satisfying (7.1).
In a more general setting we prove the following. (ii) M is diffeomorphic to X. In particular, X is a manifold.
Proof. We apply the Theorem 9.1 to obtain (after taking a subsequence if necessary) a limit solution (M, g(t), y) t∈(0,T ) := lim i→∞ (M i , i g(t), x i ) t∈(0,T ) satisfying the estimates (7.1). We prove that (M, d(g(t)), y) → (X, d X , x) as t → 0 as follows. We introduce the notation d(t) = d(g(t)) and d i (t) = d(g i (t)). In view of the Lemma 6.1 we have d GH ((B r (x i ), d i (t)), (B r (x i ), d i (0))) ≤ c(r, t) where c(r, t) → 0 as t → 0 and c(r, t) does not depend on i. Furthermore d GH ((B r (x i ), d i (0)), (B r (x), d X )) ≤ l(i, r) where l(i, r) → 0 as i → ∞, and d GH ((B r (x i ), d i (t)), (B r (y), d(t))) ≤ s(i, r, t) where s(i, r, t) → 0 as i → ∞, in view of the fact that (M, d i (t), x i ) → (M, d(t), y) and (M i , d i (0), x i ) → (X, d X , x) in GH sense as i → ∞. Hence, since Gromov Hausdorff distance satisfies the triangle inequality, we obtain (for r fixed): Letting i → ∞ (t and r fixed), we get in view of the properties of s(i, r, t) and l(i, r), and hence (M, d(g(t)), y) → (X, d X , x) as t → 0 since c(r, t) → 0 as t → 0.
Finally we show that (M, d(t), y) is diffeomorphic to (X, d, x) for all t ∈ (0, T ). The limit solution satisfies the estimates (7.1). So d(t i )(p, q) is a Cauchy sequence in i for any sequence t i → 0. In particular, we obtain a limit as i → ∞: let us call this limit l(p, q). Clearly l(p, q) does not depend on the sequence t i we choose. l(·, ·) satisfies the triangle inequality, as d(t)(·, ·) does, for all t > 0. Also l(p, p) = lim t→0 d(p, p, t) = 0. Furthermore: l(p, q) > 0 for all p = q: That is, l is a metric. From the above estimates (9.1), we see that d(t)(·, ·) → l(·, ·) as t → 0 uniformly on compact sets K ⊂ M (compact with respect to d(t) for any t) . This implies that (M, d(t), y) → (M, l, y) as t → 0 in the C 0 sense on compact sets. Now we show that the metric l defined on the set M defines the same topology as that of (M, d(t)) for any t. First note that all of the (M, d(t)) for t > 0 have the same topology: (M, g(t)) are smooth Riemannian metrics with bounded curvature evolving by Ricci flow and are all equivalent. Let us denote this topology by O. We denote the topology coming from (M, l) byÕ.
We use the notation l B r (x) to denote a ball of radius r > 0 at x ∈ M with respect to the metric l, and (as usual) B r (x) to denote a Ball of radius r > 0 at x ∈ M with respect to the metric d(t). From the above inequalities and the definition of l we have For any open set U inÕ we therefore have where r(p) > 0 is chosen small so that l B r(p) (p) ⊂ U and t(p) > 0 is chosen small so that r(p) − c 2 t(p) > 0. Hence, U is in O. Now assume V ∈ O. Then, using the estimate 9.2 again, we see that where r(p, t) is chosen small so that d(t) B r(p,t) (p) ⊂ V. Hence V ∈Õ. Hence, the identity from (M, l, y) to (M, d(t), y) is a homeomorphism. We already showed that (M, d(t), y) → (X, d X , x) as t → 0 ((X, d X , x) was defined by (X, d X , x) := lim i→∞ (M i , d i (0), x i )). Hence (X, d X , x) = (M, l, y), and (X, d X , x) is homeomorphic to (M, d(t), y). In three dimensions every manifold has a unique smooth maximal structure. This finishes the proof.
We formulate the last result of the theorem above in a form independent of the Ricci flow.
Proposition 9.3. Let (M i , g i ) be a sequence of smooth 3-manifolds (2-manifolds) in T (3, k, m, v 0 ) (T (2, k, m, v 0 )) and (X, d X , x) be a Gromov-Hausdorff limit of (M i , d(g i ), x i ) (such a (X, d X , x) always exists after taking a subsequence). Then • (X, d X ) is a manifold.
• if diam (M i , g i ) ≤ d 0 < ∞ for all i ∈ N , then M i is diffeomorphic to X for i ∈ N sufficiently large.
As a corollary to this result and Theorem 9.2 and Lemma 5.1 we obtain the following corollary. Let (X, d X ) = GH lim i→∞ (M i , d( i g 0 )). Then the solution (M, g(t), x) t∈(0,T ) obtained in Theorem 9.1 satisfies Ricci(g(t)) ≥ 0 for all t ∈ (0, T ) and (X, d X ) is diffeomorphic to (M, g(t)) for all t ∈ (0, T ). In particular, combining this with the results of Shi [46] and Hamilton [24], we get that (X, d X ) is diffeomorphic to R 3 , S 2 × R or S 3 modulo a group of fixed point free isometries in the standard metric.  [17] for an explanation. Let X q ∈ T q M be normal to γ. It is well known (see [44] chapter 1) that ρ is differentiable onM , and that ∇ρ(q) = γ ′ (l) and ∇ 2 ρ(q)(X q , X q ) = l 0 g |X ′ (s)| 2 − Riem(g)(X, γ ′ ,X, γ ′ )ds, (10.1) whereX ∈ T γ M is the unique Jacobi field along γ such thatX(0) = 0 andX(l) = X q (see the Book of do Carmo [17] for a discussion on Jacobi fields).

Hessian comparison principles
The tensor inequality ∇ 2 ρ ≤ c(n, k)g in the case that the sectional curvatures are bounded from below by k is well known: a proof may be found in (for example) [44], chapter 1. Here we show how to obtain a more general inequality which bounds ∇ 2 ρ from above and below, for constants which depend on the suprmemum of the curvatures in a geodesic Ball fo radius r where we are evaluating ∇ 2 ρ. NOTE: To be consistent with the rest of this paper I am using the convention that sectional curvatures of a plane spanned by two perpendicular vectors v, w of length one is sec(v, w) = Riem(v, w, v, w) and that the sectional curvature on the sphere is positive (in [44] sec(v, w) = Riem(v, w, w, v) > 0 on the sphere).X is a Jacobi field means then thatX ′′ − Riem(X, γ ′ , γ ′ ) = 0. Let E i ∈ T γM, i = 1, . . . , n be parallel fields (E ′ i = 0) such that {E i (t)} n i=1 is an orthonormal basis at γ(t) for for every q ∈ B r (p) ∩M as required. The estimate ∇ 2 ρ(q) ≤ R B (l) follows by using the standard hessian comparison principle (see Chapter 1 [44]), and the fact that sec(x) ≥ − sup B l (x0) |Riem(x 0 )| for x ∈ B l (x 0 )).

Estimates on the distance function for Riemannian manifolds evolving by Ricci flow
For completeness, we prove some results which are implied or proved in [28] and stated in [6] as editors' note 24 from the same paper in that book. The lemma we wish to prove is