On the regularity of Ricci flows coming out of metric spaces

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)), x_0) \to (M,d_0, x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g(t)}(x_0,1) \Subset M$ for all $t\in (0,T)$ and $d_0$ is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution $\tilde g(t)_{t\in (0,T)}$ to the $\delta$-Ricci-DeTurck flow on an Euclidean ball ${\mathbb B}_{r}(p_0) \subset {\mathbb R}^n$, which can be extended to a smooth solution defined for $t \in [0,T)$. We further show, that this implies that the original solution $g$ can be extended to a smooth solution on $B_{d_0}(x_0,r/2)$ for $t\in [0,T)$, in view of the method of Hamilton.


Overview
In this paper, we investigate and answer in certain cases the following question. and for which (M, d(g(t))) Gromov-Hausdorff converges to a metric space (M, d 0 ) as t ց 0. The Riemannian manifolds (M, g(t)) are not a priori assumed to be complete for each t ∈ (0, T ).
What further assumptions on the regularity of (M, d 0 ) and (M, g(t)) t∈(0,T ) guarantee that (M, g(t)) converges smoothly (or continuously) to a smooth (or continuous metric) as t approaches zero?
If we assume further that (M, g(t)) t∈(0,T ) is complete for each t ∈ (0, T ) and that the Ricci curvature of the solution is uniformly bounded from below, that is we assume (1.1) Ric(·, t) ≥ −1, |Rm(·, t)| ≤ c 0 t , then the existence of a unique metric d 0 whose topology agrees with that of (M, g(t)) t ∈ (0, T ) is guaranteed by the results in [17,19]. More explicitly, defining d t = d(g(t)) to be the distance with respect to g(t) on M , in Lemma 3.1 of [19] it is shown that d t → d 0 for a metric d 0 on M and that the estimates hold in this case, which implies convergence of d t in the C 0 sense to d 0 , and is a stronger convergence than that of Gromov-Hausdorff convergence. In particular, this implies that the metric space (M, d 0 ) has the same topology as (M, d(g(t))) for all t ∈ (0, T ), as is shown for example in the proof of Theorem 9.2 of [17], or can be seen directly using the estimate (3.4) of [19]. Since Gromov-Hausdorff limits are unique up to isometries, we see that the following is true in this setting: if (M, d(g(t i )), p) → (X, d X , y) in the Gromov-Hausdorff sense for a sequence of times t i > 0 with t i ց 0 as i → ∞, then (X, d X , y) is isometric to (M, d 0 , p) which is a manifold. Hence it is not possible that complete solutions satisfying (1.1) come out of metric spaces which are not manifolds. Note that if we have Ric(·, t) ≥ −C, and |Rm(·, t)| ≤ c 0 /t for all t ∈ (0, T ) for some C ≥ 1, then if we scale once,g(t) = Cg(t/C), then the new solutiong(·,t) satisfies (1.1).
That is, assuming B g(t) (x 0 , ℓ) ⋐ M for all t ∈ (0, T ) for some ℓ > 0, we can always assume that the metric d 0 exists (locally) and that the convergence is in the above sense, (1.2) (which is stronger than Gromov-Hausdorff convergence) when examining the local behaviour of the metric near a point x 0 ∈ M .
Examples of solutions satisfying the conditions (1.1) are as follows: 1. Expanding gradient Ricci solitons coming out of smooth cones (M n , d X ) where (M n , d X ) is the completion of (R + × S n−1 , dr 2 ⊕ r 2 γ), where γ is a Riemannian metric on the sphere, which is smooth and whose curvature operator has eigenvalues larger than one. 'Coming out' here means that (M, d(g(t))) → (M n , d X ) in the pointed Gromov-Hausdorff sense as t → 0. In the paper [14] of the last two named authors of the current paper, cones of this type which arise as smooth limits (away from the tip) of a blow-down of a non-compact manifold with non-negative, bounded curvature operator and asymptotically Euclidean volume growth were considered. They showed, in this case, that there is an expanding soliton coming out of the cone, with non-negative curvature operator, satisfying (1.1).
The construction of the solution in the paper [14] guarantees, that the convergence as t → 0 to the initial values is, in this case, in the C ∞ loc sense away from the tip. Later, the first named author of the current paper, Deruelle, extended these results in [8] to show that there is always a soliton coming out of any smooth cone of the type considered at the beginning of this example. The construction of Deruelle also guaranteed that the convergence is in the C ∞ loc sense away from the tip: this existence result is based on the Nash-Moser fixed point theorem. Problem 1.1 was partly motivated by the cost of using such a "black box". Indeed, the Nash-Moser fixed point theorem is not so sensitive to the nature of the non-linearities of the Ricci flow equation as long as the corresponding linearized operator satisfies the appropriate Fredholm properties. In particular, the use of the Nash-Moser fixed point theorem does not shed new light on the smoothing effect of the Ricci flow. Finally, we emphasize the fact that uniqueness of such solutions is unknown among the class of asymptotically conical gradient Ricci solitons with positive curvature operator. Indeed, the proof given in [8] only works if any such two expanding gradient Ricci solitons with the same tangent cone at infinity have a vanishing Bianchi gauge.
2. Let (M n i (0), g i (0), x i ) i∈N be a sequence of smooth Riemannian manifolds with bounded curvature, such that R(g i (0)) + c · Id(g i (0)) ∈ C K and Vol(B gi(0) (x)) ≥ v 0 for all x ∈ M i for all i ∈ N, for some c, v 0 > 0 where R is the curvature operator, Id is the identity operator of the sphere, Id(g) ijkl = g ik g jl − g jk g il , and C K is the cone of i) non-negative curvature operators, respectively ii) 2-non-negative curvature operators, respectively iii) weakly P IC 1 curvature operators, respectively iv) weakly P IC 2 curvature operators. Then [ [17] for (i), (ii) in case n = 3, [2] for (i) -(iv) for general n ∈ N] there are solutions (M i , g i (t), x i ) t∈[0,T (n,v0)] such that R(g(t))) + C · Id ∈ C K (for some new C > 0) which is stronger than, i.e. it implies, the condition Ric(g(t)) ≥ −c(n)C. After scaling each solution once by a large constant K = K(n, v 0 ),g i (t) = Kg i (t/K), we obtain a sequence of solutions satisfying (1.1). After taking a sub-sequence, we obtain a pointed Cheeger-Hamilton limit solution (M n , g(t), x ∞ ) t∈(0,1) which satisfies (1.1).
Problem 1.1 can be considered locally in the context of the above examples as follows.
In the setting of Example 3) let (M, d 0 , x ∞ ) be the limit as t ց 0 of (M, d(g(t)), x ∞ ). We note that (M, d 0 , x ∞ ) is also isometric to the Gromov-Hausdorff limit of (M i , d(g i (0)), x i ) as i → ∞, in view of the estimates (1.2). Let V ⊆ M be an open set such that (V, d 0 ) is isometric to a smooth (continuous) Riemannian manifold. Can (V, g(t)) t∈(0,1) be extended smoothly (continuously) to t = 0, that is, does there exist a smooth (continuous) g 0 on V such that (V, g(t)) t∈[0,1) is smooth (continuous)?
We will see in Theorem 1.3 in the next section, that the answer to each of these questions in the smooth setting is yes, if we measure the smoothness of the initial metric space appropriately. The answer to each of these questions is also yes in the continuous setting, see Theorem 1.4 in the next section, if we measure the continuity of the initial metric appropriately and the convergence in the continuous setting is measured up to diffeomorphisms.
The smoothness (respectively continuity) of a metric space in this paper will be measured in the following way. In the following 0 < ε 0 (n) < 1 is a small fixed positive constant depending only on n. We denote with B r (x) ⊂ R n the Euclidean ball with radius r, centred at x. Definition 1.2. Let (X, d 0 ) be a metric space and let V be a set in X. We say (V, d 0 ) is smoothly (respectively continuously) n-Riemannian if for all x 0 ∈ V there exist 0 <r, r withr < 1 5 r and points a 1 , . . . , a n ∈ B d0 (x 0 , r) such that the map is a (1 + ε 0 ) Bi-Lipschitz homeomorphism on B d0 (x 0 , 5r) and the push-forward of d 0 by F 0 denoted byd 0 and defined byd 0 (x,ỹ) : 5r)) is induced by a smooth Riemannian metric, that is, more specifically, there exists a smooth (respectively continuous) Riemannian metricg 0 defined on B 4r (F 0 (x 0 )), such thatd 0 satisfiesd 0 = d(g 0 ), when restricted to Br(F 0 (x 0 )).
For a Riemannian metric g defined on an open set Ω in R n , we have used the notation d(g) to refer to the metric on Ω defined by d(g)(x, y) = inf{ L g (γ) | γ a smooth path in Ω from x to y}, for all x, y ∈ Ω, where L g (γ) is the length of the curve γ with respect to g.

Main results
The first theorem stated here gives a positive answer to the questions posed in the smooth setting in the previous subsection. Theorem 1.3. Let (M, g(t)) t∈(0,T ] be a smooth solution to Ricci flow satisfying (1.1) and assume B g(t) (x 0 , 1) ⋐ M for all 0 < t ≤ T and let (B d0 (x 0 , r), d 0 ) be the C 0 limit of (B g(t) (x 0 , r), d(g(t))) as t ց 0 (which always exists in view of [19,Lemma 3.1]). Assume further that (B d0 (x 0 , r), d 0 ) is smoothly n-Riemannian in the sense of Definition 1.2 near x 0 . Then there exists a smooth Riemannian metric g 0 on B d0 (x 0 , s) for some s > 0 such that we can extend the smooth solution (B d0 (x 0 , s), g(t)) t∈(0,T ) to a smooth solution (B d0 (x 0 , s), g(t)) t∈[0,T ) by defining g(0) = g 0 .
As noted in the paper [20] by Topping, this result was known to be correct in dimension two for compact manifolds without boundary, by results of Richard in [13].
The next theorem is concerned with the questions asked in the previous subsection in the continuous setting.
) t∈(0,T ] be a smooth solution to Ricci flow satisfying (1.1) and assume B g(t) (x 0 , 1) ⋐ M for t ≤ T and let (B d0 (x 0 , r), d 0 ) be the C 0 limit of (B g(t) (x 0 , r), d(g(t))) as t ց 0 (which always exists in view of [19,Lemma 3.1]). Assume further that (B d0 (x 0 , r), d 0 ) is continuously n-Riemannian in the sense of Definition 1.2. Then for any sequence t i with 0 < t i → 0 as i → ∞, there exists a radius v > 0 and a continuous Riemannian metricg 0 , defined on B v (p), p ∈ R n , and family of smooth

Metric space convergence and the condition (1.1)
Assume we have a smooth complete solution to Ricci flow (M n , g(t)) t∈(0,1) , satisfying |Rm(·, t)| ≤ c 0 /t for all t ∈ (0, 1) for some c 0 ≥ 1, but that we don't assume the condition Ric(g(t)) ≥ −k for all t ∈ (0, 1) for some k ∈ R + . Then there is no guarantee that a limit metric d 0 = lim t→0 d(g(t)) exists. If we have a sequence of smooth complete solutions (M n i , g i (t), x i ) t∈[0,1) , satisfying |Rm(g i (t)))| ≤ c 0 /t and Vol(B gi(t) (x)) ≥ v 0 > 0 for all t ∈ (0, 1) for all x ∈ M i for some c 0 , v 0 > 0, for all i ∈ N, we obtain a limiting solution in the smooth Cheeger-Hamilton sense, (M n , g(t), p) t∈(0,1) , which satisfies |Rm(·, t)| ≤ c 0 /t for all t ∈ (0, 1), but again, there is no guarantee that a limit metric d 0 = lim t→0 d(g(t)) exists. Furthermore, if a pointed Gromov-Hausdorff limit (M, d 0 , p), as t → 0, of (M, d(g(t)), p) exists and if a Gromov-Hausdorff limit (X, d X , y) in i ∈ N of (M n i , d(g i (0)), x i ) i∈N exists, then there is no guarantee that (X, d X , y) is isometric to (M, d 0 , p), or that (M, d 0 ) has the same topology as d(g(t)) for t > 0.
An example which considers the metric behaviour under limits of solutions with no uniform bound from below on the Ricci curvature but with |Rm(·, t)| ≤ c 0 /t is given in a recent work of Peter Topping [20]. There, he constructs examples of smooth solutions (T 2 , g i (t)) t∈[0,1) to Ricci flow, satisfying (T 2 , d(g i (0))) → (T 2 , d(δ)), as i → ∞, where δ is the standard flat metric on T 2 , and |Rm(g i (t))| ≤ c/t for all t ∈ (0, 1), i ∈ N for some c > 0, but so that the limiting solution (T 2 , g(t)) t∈(0,1] satisfies (T 2 , d(g(t))) t∈(0,1) = (T 2 ,d), where (T 2 ,d) is isometric to (T 2 , d(2δ)). The initial smooth data g i (0) do not satisfy Ric(g i (0)) ≥ −k for some fixed k > 0 for all i ∈ N, and so the arguments used to show that the Gromov-Hausdorff limit of the initial data is the same as the limit as t → 0 of the limiting solution, are not valid.

Related results and works
We begin by considering the heat equation. If we have a smooth solution u : R n ×(0, 1) → R, with ∂ ∂t u = ∆u, and we know that u(·, t) → u 0 (·) locally uniformly on B 1 (0), where u 0 is smooth on B 1 (0), then the solution can be locally extended to a smooth local solution v : B 1/2 (0) × [0, 1) → R, by defining v(·, 0) = u 0 (·) on B 1/2 (0), as we now explain. We know that the function u : B 3/4 (0) × [0, 1] → R is continuous. Hence using the linear parabolic theory, see for example Theorem 2.1 of this paper with M = R n and g(t) = δ for all t ∈ [0, 1], we can find a map z : 1]. From the maximum principle, we see that z − u = 0 and hence u = z is smooth on B 3/4 (0) × [0, 1] as required. Here the linear theory simplifies the situation. We have also assumed that u(·, t) → u 0 locally uniformly. In the Ricci flow setting, assuming (1.1), we saw above that the initial values must be taken on uniformly, albeit for the distance, not necessarily the Riemannian metric.
A non-linear setting closer to the one we consider is as follows. In the paper [1], Appleton considers (among other things) the δ-Ricci-DeTurck flow of metrics g 0 on R n which are close to the standard metric δ, in the sense that |g 0 − δ| δ ≤ ε(n). In the work of Koch and Lamm, see [11,Theorem 4.3], it was shown that there always exists a weak solution (R n , g(t)) t∈(0,∞) in this case. Weak solutions defined on [0, T ) (T = ∞ is allowed) are smooth for all t > 0 and h(x, t) := g(x, t) − δ(x) has bounded X T norm, where If the initial values g 0 are C 0 then the initial values are realised in the C 0 sense, that is |g(t) − g 0 | δ → 0 as t → 0. Appleton showed, see [1,Theorem 4.5], that any weak solution h(t) := g(t) − δ which has g 0 ∈ C 2,α loc (R n ) and |h 0 | δ ≤ ε(n), must have h(t) ∈ H 2+α,1+ α 2 loc (R n × [0, ∞)). In particular the zeroth, first and second spatial derivatives of h(t) locally approach those of h 0 as t ց 0. That is, for classical initial data h 0 ∈ C 2,α loc (R n ), any weak solution h(t) restricted to Ω approaches h(0) in the C 2,α (Ω) norm on Ω for any precompact, open set Ω. Raphaël Hochard has in his PhD-thesis [10] proven some results similar to, or the same as, some of those appearing in Sections 2 and 3 of this paper. We received a copy of Hochard's thesis, after a pre-print version, including the relevant sections, of this paper was finished but not yet published. We have included references to the results of Hochard at the appropriate points throughout this paper. His approach, in studying the relevant objects, differs slightly, as we explain at the relevant points.

Outline of Paper
We outline the proof method of the main theorems, Theorem 1.3 and 1.4. The method is somewhat similar to the one we used above to show smoothness of solutions to the heat equation coming out of smooth initial data, which are a priori smooth for times larger than zero.
We show, in the setting we are considering, that there is a smooth, on the interior, solution to the Ricci-harmonic map heat flow, Z : B d0 (x 0 , 1) × [0, T ) → R n , with initial and boundary values given by the map F 0 , which represents distance coordinates at time zero. The a priori estimates we prove in Sections 2 and 3 help us to construct the solution, and from the Regularity Theorem 3.7, we see that the solution is 1 + α(n) Bi-Lipschitz at any time t ∈ (0, S(n)) ∩ [0, T 2 ). The explicit construction of Z is carried out in Theorem 3.10.
Hence, we may consider the push forwardg(t) := (Z t ) * (g(t)), which is then, by construction a solution to the δ-Ricci-DeTurck flow with background metric δ, andg(t) is α(n) close to δ in the C 0 sense. We then restrict to the case that the push forward of d 0 with respect to F 0 is generated locally by a continuous Riemannian metricg 0 . A further application of the regularity theorem, Theorem 3.7 of Section 3, then shows thatg(t) converges to the continuous metricg 0 (locally). This is explained in detail in Theorem 4.3 in Section 4. If we assume further thatg 0 is smooth, and sufficiently close to δ, then we consider the Dirichlet Solution ℓ to the δ-Ricci-DeTurck flow on an Euclidean ball B r (0) × [0, T ], with parabolic boundary data given byg. The existence of this solution is shown in Section 5, where Dirichlet solutions to the δ-Ricci-DeTurck flow with given parabolic boundary values C 0 close to δ are constructed. The L 2 -lemma, Lemma 6.1 of Section 6, tells us that the (weighted) spatial L 2 norm of the difference g 1 − g 2 of two solutions g 1 , g 2 to the δ-Ricci-DeTurck flow defined on an Euclidean ball is non-increasing, if g 1 and g 2 have the same values on the boundary of that ball, and are sufficiently close to δ for all time t ∈ [0, T ]. An application of the L 2 -lemma then proves that ℓ =g. The construction of ℓ, carried out in Section 5, then guarantees that ℓ is smooth on B r (0) × [0, T ]. Henceg is smooth on B r (0) × [0, T ]. Section 7 completes the proof of Theorem 1.3: The smoothness ofg on B r (0) × [0, T ] implies that one can extend g smoothly (locally) to t = 0. In Section 7 we also discuss some of the consequences of Theorem 1.3 in the context of expanding gradient Ricci solitons with non-negative Ricci curvature.

An open problem
The lower bound on the Ricci curvature in (1.1) is used crucially to obtain the bound from above for d t in (1.2). It is also used in Section 4, when showing thatg(t) converges tog 0 in the C 0 norm. Problem 1.5. Can the bound from below on the Ricci curvature in Section 3 and/or other sections be replaced by a weaker condition?
We comment on this at various points in the paper.
(1c) If g is locally in C 2 : Ric(g) is the Ricci Tensor, Rm(g) is the Riemannian curvature tensor, and R(g) is the scalar curvature.
(2) For a one parameter family (g(t)) t∈(0,T ) of Riemannian metrics on a manifold M , the distance induced by the metric g(t) is denoted either by d(g(t)) or d t for t ∈ (0, T ).
(4) B R (m) always refers to an Euclidean ball with radius R > 0 and centre m ∈ R n .

Acknowledgements
The first author is supported by grant ANR-17-CE40-0034 of the French National Research Agency ANR (Project CCEM) and Fondation Louis D., Project "Jeunes Géomètres". The third author is supported by the SPP 2026 'Geometry at Infinity' of the German Research Foundation (DFG).
2 Ricci-harmonic map heat flow for functions with bounded gradient We prove some local results about the Ricci-harmonic map heat flow. R. Hochard, in independent work, proved some results in his Ph.D.-thesis which are similar to or the same as those of this chapter, see [10, Section II.3.2]. Hochard uses blow up arguments (i.e. contradiction arguments combined with scaling arguments) to prove some of his estimates, whereas we use a more direct argument involving the maximum principle applied to various evolving quantities. The first theorem we present is a local version of a theorem of Hamilton, [9, p. 15] in the setting that the curvature of the Ricci flow is bounded by a constant times the inverse of time, and the Ricci curvature is bounded from below.
Theorem 2.1. Let (M n , g(t)) t∈[0,T ) be a smooth solution to Ricci flow with Then there is a unique solution to the Dirichlet problem, for the Ricci-harmonic map heat flow
Proof. Since B g(0) (x 0 , 1) ⊂ M is a compact set, and the solution (M, g(t)) t∈[0,T ) is smooth, we see that we can find a finite collection of coordinate charts which cover B g(0) (x 0 , 1) such that the metric is uniformly equivalent to the standard metric δ (in these coordinates) and that all derivatives of g (in space and time) of any order are uniformly bounded in these coordinates.
Hence there is a unique solution to the Dirichlet problem (2.2). For the reader's convenience we give a brief explanation of why this is the case. We modify the boundary data slightly, in order to satisfy the compatibility conditions of k-th order, i.e. we set where ϕ i := ∂ i t | t=0 F is the right-hand side we obtain if we differentiate the equation where P is the parabolic boundary P : This problem satisfies the compatibility conditions of k-th order at t = 0. Furthermore, notice thatF α (·, t) = F 0 (·) for t ≥ 2α, and F α (·, 0) = F 0 (·).
Next, we consider the related problem of finding a solution Now S α satisfies the compatibility conditions at least of order 1 at time t = 0 (which is equivalent to Z α (·, 0) = 0 on the boundary of B g(0) (x 0 , 1) in this case, since the parabolic boundary data is zero), and so we find a solution for all t ≤ T , and so [12, Lemma 6.1, Chapter VII] is applicable, which then tells us that |∇S α || P ≤ c(g| [0,T ] , F 0 , T ).
The maximum principle applied to |∇S α | then shows us that |∇S α | ≤ c(g| [0,T ] , F 0 , T ). Hence, |S α || Pε ≤ c(ε) where P ε consists of the points which are in an ε-tubular neighbourhood of P measured with respect to the parabolic distance and where here c(ε) → 0 as ε → 0. Going back to H α , that is setting

Interior estimates applied to the evolution equation for
: a maximum principle with cut off and the difference quotient method suffices to show this. By Arzelá-Ascoli's Theorem, we now get the required solution by taking a limit as α goes to 0.
This finishes the brief explanation of the existence result. We now return to the proof of Theorem 2.1.
The statement (2.3) follows from the Maximum Principle and the evolution equation for Statement (2.4) follows from the distance estimates, (1.2), which hold on B g(0) (x 0 , 1) for any solution to Ricci flow satisfying (2.1) and Regarding (2.5), we first recall the following fundamental evolution equation satisfied by |∇F | 2 g(t) : Notice that the term Ric(g(t))(∇F, ∇F ) showing up in the Bochner formula applied to ∇F cancels with the pointwise evolution equation of the squared norm of ∇F along the Ricci flow.
In case the underlying manifold is closed, the use of the maximum principle would give us the expected result.
We consider the function Z := η|∇F | 2 g + c 2 |F | 2 , with c 2 = 10c(n)c 3 (n). The quantity Z is less than c 1 + 4c 2 everywhere at time zero. We consider a first time and point where Z becomes equal to c 1 + 5c 2 on B g(0) (x 0 , 1). This must happen in B g(0) (x 0 , 1), since η = 0 on a small open set U containing ∂B g(0) (x 0 , 1) and c 2 |F | 2 < 4c 2 by (2.3). At such a point and time, we have by (2.6) and (2.7) together with the properties of η, by the choice of c 2 , which is a contradiction. Hence Z(x, t) ≤ c 1 + 5c 2 for all t ≤ S(n, c 0 ), which implies We want also to prove an estimate for the second covariant derivatives of a solution to the Ricci-harmonic map flow. In fact, once we have a solution to the Ricci-harmonic map heat flow with bounded gradient, the solution smooths out the second derivatives in a controlled way, as the following theorem shows.
3. For all c 1 > 0 and n ∈ N, there exists a positive numberε 0 (c 1 , n) such that the following is true. Let (M n , g(t)) t∈[0,T ) be a smooth solution to Ricci flow such that is sufficiently small is not necessary : |Rm(·, t)| ≤ k/t with k arbitrary is sufficient in our argument, as can be seen by examining the proof, but then the times t for which the conclusions are valid, are required to satisfy t ≤ S(c 1 , k, n), where S(c 1 , k, n) > 0 is sufficiently small. We only consider small k, as this is sufficient for the setting of the following chapters. A version of this theorem, with c 1 = c(n) and the condition |Rm(·, t)| ≤ k/t, k arbitrary, was independently proved by Raphaël Hochard in his Ph.D.-thesis using a contradiction argument: see [10, Lemma II.3.9].
Proof. In the following, we denote constants C(ε 0 , c 1 , n) simply by ε 0 if C(ε 0 , c 1 , n) goes to 0 as ε 0 tends to 0 and c 1 and n remain fixed. For example c 2 1 n 4 ε 0 and b(c 1 , n) √ ε 0 are denoted by ε 0 if b(c 1 , n) is a constant depending on c 1 and n. Let . The evolution equation for Z can be calculated as follows. Locally, where ∂ i stands for a partial derivative. Shi's estimates and the distance estimates (1.2) guarantee that |∇Rm(g(t))| ≤ ε 0 t −3/2 for t ≤ S(n) on B g(t) (x 0 , 3/4) : see for example Lemma 3.1 (after scaling once by 400). For the sake of clarity in the computation to follow, we use the notation ∇ to denote ∇ g(t) at a time t, Rm to denote Rm(g(t)) at a time t, et cetera, although the objects in question do indeed depend on the evolving metric g.
Here we have used freely the formula for the commutation of the second covariant derivatives of a tensor T , : see for example [21]. This means in particular that, . (2.8) Using (2.7) together with (2.8), one sees that the following evolution (in)equalities are satisfied by Z: where we assume that a 0 ≥ 1, c(n, c 1 ) denotes a positive constant depending on the dimension n and the Lipschitz constant c 1 , that may vary from line to line and we have used Young's inequality freely. Therefore, the function Z satisfies the following: if a 0 is chosen sufficiently large such that a 0 ≥ c(n, c 1 ).
In case the underlying manifold is closed, the use of the maximum principle would give us the expected result: if there is a first time and point (x, t) where Z(x, t) = 10a 2 0 for example, we obtain a contradiction. Hence we must have Z ≤ 10a 2 0 .
In order to localize this argument, we introduce, as in the proof of Theorem 2.1, a Perelman type cut-off function η : η(·, t) everywhere, and |∇η| 2 g(t) ≤ c(n)η everywhere, as long as t ≤ S(n) ≤ 1: see for example [18,Section 7] for details.
We first derive the evolution equation of the functionẐ := ηZ with the help of inequality (2.9) for t ≤ S(n): Multiplying the previous differential inequality (2.10) by the non-negative function η gives: where again, c(n) denotes a positive constant depending on the dimension only and which may vary from line to line. If the maximum ofẐ at any time is larger that 100a 2 0 then this value must be achieved at some first time and point (x, t) with t > 0 , sinceẐ(·, 0) = 0. This leads to a contradiction if t ≤ S(n) ≤ 1.

Almost isometries, distance coordinates and Ricciharmonic map heat flow
In this section we are interested in smooth, not necessarily complete, solutions to Ricci flow (M, g(t)) t∈(0,T ) with T ≤ 1 where M is connected, satisfying the following: given where γ is any smooth curve between x and y in M and L g(t) is the length of this curve with respect to g(t).
By scaling the solution once by ε 2 0 /R 2 ≥ 1 we may assume without loss of generality that the inequalities (3.1) and inclusion (3.2) above hold for all 0 ≤ r < t, s ∈ [0, 1] ∩ (0, T ). We assume this in the following.
Also the estimates of Shi hold, as is explained in the following lemma.
In the main theorem of this chapter, Theorem 3.7, we consider maps F t , and ε 0 < 1 which satisfy the following property: If such a map F t exists for all t ∈ (0, T ), and we further assume that sup t∈(0,T ) |F t (x 0 )| < ∞, then for any sequence t i > 0, t i → 0 with i → ∞, we can, after taking a subsequence, find a limiting map, F 0 which is the using a diagonal subsequence and the theorem of Heine-Borel, and then we extend F 0 uniquely, continuously to all of B d0 (x 0 , 100), which is possible in view of the fact that the Bi-Lipschitz property (3.4) is satisfied on D. The sequence (F ti ) i converges uniformly to F 0 in view of (3.1), (c) and (3.4).
In Theorem 3.7, we see that if we consider a Ricci-harmonic map heat flow of one of the functions F t (for small enough t), and we assume that the solution satisfies a gradient bound, |∇F t | g(t) ≤ c 1 , on some ball, then after flowing for a time t, the resulting map will be a 1 + α 0 Bi-Lipschitz map on a smaller ball, where α 0 if ε 0 ≤ε 0 (n, α 0 , c 1 ) is small enough. This property continues to hold if we flow for a time s where t ≤ s ≤ min(S(n, α 0 , c 1 ), T ).
For convenience, we introduce the following notation: for all x, y ∈ W .
As a consequence of this assumption and the distance estimates (3.1), we see that the corresponding distance coordinates at time t, F t : B d0 (x 0 , 50) → R n , given by F t (x) := (d t (a 1 , ·), . . . , d t (a n , ·)), are mappings satisfying property (c), for all t ∈ (0, T ). Remark 3.3. R. Hochard also looked independently at some related objects in his Ph.D.-thesis, and some of the infinitesimal results he obtained there are similar to those of this section, c.f. [10, Theorem II.3.10], as we explained in the introduction. Hochard considers points x 0 which are so called (m, ε) explosions at all scales less then R (only m = n is relevant in this discussion). The condition, for m = n, says that there exist points p 1 , . . . , p n such that for all x in the ball B d0 (x 0 , R) and all r < R, there exists an εr GH approximation ψ : B d0 (x, r) → R n such that the components ψ i are each close to the components of distance coordinates d(·, p i ) − d(x, p i ) at the scale r, in the sense that |ψ i (·)−(d(·, p i )−d(x, p i ))| ≤ εr on B d0 (x, r). Our approach and our main conclusion differ slightly to the approach and main conclusions of Hochard. The condition (c) we consider above looks at the closeness of the maps F t to being a Bi-Lipschitz homeomorphism, and this closeness is measured at time t using the maps F t , and our main conclusion, is that the map will be a 1 + α 0 Bi-Lipschitz homeomorphism after flowing for an appropriate time by Ricci-harmonic map heat flow, if t > 0 is small enough. We make the assumption on the evolving curvature, that it is close to that of R n , after scaling in time appropriately. Nevertheless, the proof of Theorem 3.7 below and of [10, Theorem II.3.10] have a number of similarities, as do some of the concepts.
The main application of this section is, assuming (a),(b), and that F 0 are distance coordinates which define a 1 + ε 0 Bi-Lipschitz homeomorphism, to show that it is possible to define a Ricci-DeTurck flow (g(s)) s∈(0,T ] starting from the metricd 0 := (F 0 ) * d 0 , on some Euclidean ball, which is obtained by pushing forward the solution (g(s)) s∈(0,T ] by diffeomorphisms. The sense in which this is to be understood, respectively this is true, will be explained in more detail in the next paragraph. The strategy we adopt is as follows. Assuming (a), (b), we consider the distance maps F ti at time t i defined above for a sequence of maps t i > 0 with t i → 0. We mollify each F ti at a very small scale, so that we are in the smooth setting, but so that the essential property, (c), of the F ti is not lost (at least up to a factor 2). Then we flow each of the F ti on B d0 (x 0 , 100) by Ricci-harmonic map heat flow, keeping the boundary values fixed. The existence of the solutions Z ti : B d0 (x 0 , 100)×[t i , T ) → R n , is guaranteed by Theorem 2.1. According to Theorem 3.7, the Z ti (s) are then 1+α 0 Bi-Lipschitz maps (on a smaller ball), for all s ∈ [2t i , S(n, ε 0 )] ∩ (0, T 2 ), if ε 0 = ε 0 (α 0 , n) is small enough. If we take the push forward of g(·) with respect to Z ti (·) (on a smaller ball), then after taking a limit of a subsequence in i, we obtain a solutiong(s) s∈(0,S(n,ε0))∩(0, T 2 ) to the δ-Ricci-DeTurck flow such that (1 − α 0 )δ ≤g(s) ≤ (1 + α 0 )δ for all s ∈ (0, S(n, ε 0 )) ∩ (0, T 2 ) in view of the estimates of Theorem 3.7. The solution then satisfies d(g(t)) →d 0 := (F 0 ) * (d 0 ) as t → 0 and hence may be thought of as a solution to Ricci-DeTurck flow coming out ofd 0 . This is explained in Theorem 3.10. In the next section we examine the regularity properties of this solution, which depend on the regularity properties ofd 0 .

Almost isometries and Ricci-harmonic map heat flow
In this subsection we provide some technical lemmas giving insight into the evolution of almost isometries under Ricci-harmonic map heat flow. These results will be needed in the following subsection.
Proof. If not, then for some σ > 0, we have a sequence of maps L i : B i −1 (0) → R n such that L i is a weak 1 i isometry, but L i is not σ close in the C 0 sense to any element S ∈ O(n) on B σ −1 (0). By taking a subsequence we obtain convergence in the C 0 sense of L i to an element S ∈ O(n), since the maps are Lipschitz, and hence we can find a subsequence which converges in the C 0,α norms on compact subsets. This leads to a contradiction.
Since |∇(F − S)| 2 + |∇ 2 (F − S)| 2 ≤ c(n)c 1 for s ∈ [1/2, 1] on B β −1 /2 (0) we can use interpolation inequalities as in [6, Lemma B.1] to deduce that on B β −1 /4 (0) Again, Since h(t) is α-close to δ this implies that for α sufficiently small for all v ∈ T y R n of length one with respect to h(s) for s ∈ [1/2, 1] and y ∈ B β −1 /4 (0). We also see that where by the mean value theorem p is some point on the unit speed line between x and y, and v p is a vector of length one with respect to δ and for all x, y ∈ B β −1 /4 (0), for all s ∈ [1/2, 1], t ∈ [0, 1] for α sufficiently small.
Lemma 3.6. For all n, k ∈ N, L > 0 there exists an ε 0 = ε 0 (n, k, L) > 0 such that the following holds. Let M n be a connected smooth manifold, and g and h be smooth and that there exists a map F : B h (y 0 , L) → R n which is an ε 0 almost isometry with respect to h, that is for all z, y ∈ B h (y 0 , L), and F (y 0 ) = 0. Then (B h (y 0 , L/2), g) is 1/L-close to the Euclidean ball (B L/2 (0), δ) in the C k -Cheeger-Gromov sense.
Proof. Assume it is not the case. Then there is an L > 0 for which the theorem fails. Then we can find sequences g(i), h(i),M (i), F (i) : B h(i) (y i , L) → R n satisfying the above conditions with ε 0 := 1/i but so that the conclusion of the theorem is not correct. Using the almost isometry, we see that for any ε > 0 we can cover B h(i) (y 0 (i), 6L/7) by N (ε) balls (with respect to h) of radius ε, for all i. Hence, using |d h(i) −d g(i) | C 0 (B h (y0,L)) ≤ 1/i, we see that the same is true for B g(i) (y 0 (i), 5L/6) ⊆ B h(i) (y 0 (i), 6L/7) with respect to g(i): we can cover B g(i) (y 0 (i), 5L/6) by N (ε) balls (with respect to g(i)) of radius ε, for all i. Hence, due to the compactness theorem of Gromov (see for example [3, Theorem 8.1.10]), there is a Gromov-Hausdorff Limit In particular, there must exist which are ε(i) Gromov-Hausdorff approximations, where ε(i) → 0 as i → ∞. Using the maps G(i) and the 1/i almost isometries F (i), we see that there is a pointwise limit map, which is an isometry. Hence the volume of B g(i) (y 0 (i), 2L/3) converges to ω n (2L/3) n (in particular the sequence is non-collapsing) as i → ∞, since volume is convergent for spaces of bounded curvature (which are for example Aleksandrov spaces and spaces with Ricci curvature bounded from below). Hence (B g(i) (y 0 (i), L/2), g(i)) converges to (B L/2 (0), δ) in the C k norm in the Cheeger-Gromov sense, which leads to a contradiction if i is large enough.
3.2 Ricci-harmonic map heat flow of (1 + ε 0 )-Bi-Lipschitz maps and distance coordinates We begin with the Regularity Theorem, Theorem 3.7, for solutions to the Ricci-harmonic map heat flow, whose initial values are sufficiently close to those of a 1 + ε 0 Bi-Lipschitz map.
Hence there are geodesic coordinates ϕ on the ball (Bd 0 (x, α −1 ),g(σ)) such that the metricg(σ) written in these coordinates is α close to δ in the C k norm, if we keep σ fixed and choose ε 0 small enough. Using (3.17), and the evolution equation ∂ ∂tg = −2Ric(g) in the coordinates ϕ, we see that the evolving metric h(·) = ϕ * (g(·)) in these coordinates is also, without loss of generality, α close to δ for t ∈ [t, 2] in the C 2 norm.
With the help of the previous theorem, we now show that it is possible to construct a solution to the δ-Ricci-DeTurck flow coming out ofd 0 := (F 0 ) * d 0 using the harmonic map heat flow, if we assume that (ĉ) is satisfied. First we show that by slightly mollifying the distance coordinates at time t, we obtain maps which satisfy (c).  a 1 , x), . . . , d t (a n , x)) using the (fixed) points a 1 , . . . , a n from (ĉ) (which define distance coordinates at t = 0). Then by slightly mollifying F t we obtain a mapF t : B d0 (m 0 , R) → R n which is smooth and satisfies |∇F t | g(t) ≤ c(n) as well as (c) (with ε 0 replaced by 2ε 0 ), provided t ≤T (ε 0 , R).
Proof. As already noted, F t | B d 0 (x0,50) : B d0 (x 0 , 50) → R n satisfies (c) in view of the distance estimates (3.1), if t ≤T (ε 0 , R). Also, it is well known, that the Lipschitz norm of any map F t as defined above may be estimated by a constant depending only on n: in view of the triangle inequality. Hence, by slightly mollifying the map F t , we obtain a mapF t : B d0 (x 0 , 50) → R n which is smooth and satisfies (c) (with ε 0 replaced by 2ε 0 ) and |∇F t | g(t) ≤ c(n). (ii) If we remove condition (ĉ) and replace it by the assumption: there exists a sequence of times t i > 0 with t i → 0 as i → ∞, and mapsF ti : B d0 (x 0 , 100) → R n each of which satisfies (c), sup i∈N |F ti (x 0 )| < ∞, and |∇F ti | g(ti) ≤ c 1 then we can use theseF ti in the above, instead of the slightly mollified distance functions, and the conclusions of the theorem still hold for s ≤Ŝ := min(S(n, c 1 , α 0 ), T /2] if the ε 0 = ε 0 (n, c 1 , α 0 ) appearing in (a) and (c) is small enough. In this case, F 0 is the uniform C 0 limit of a subsequence of the F ti as i → ∞ and satisfies (3.4). The existence of such an F 0 is always guaranteed in this setting, as explained directly after the introduction of the condition (c).
Proof. Theorem 3.7 tells us that the maps Z ti (s) : ). This is well defined in view of Theorem 3.7. Theng i is a solution to the δ-Ricci-DeTurck flow on B 3/2 (F ti (x 0 )) (see [9,Chapter 6] for instance) and satisfies the metric inequalities (3.18) for all s ∈ (2t i ,Ŝ) in view of Corollary 3.8. Using [16,Lemma 4.2] we see that (0)). Taking a subsequence in i we obtain the desired solutiong(s) s∈(0,Ŝ) on B 1 (F 0 (0)) with . The Z ti all satisfy the estimates stated in the conclusions of Theorem 3.7, and so there is a uniform C 1,α limit map Z : B d0 (x 0 , 2) × (0,Ŝ) → R n , in view of the Theorem of Arzelà-Ascoli. Furthermore, |Z(·, s) − F 0 (·)| ≤ c 1 √ s for s ∈ (0,Ŝ) in view of the estimate (3.8). Let v, w ∈ B 1 (0) be arbitrary, and x, y the corresponding points in B d0 (x 0 , 2) at time s, that is the unique points x, y with Z s (x) = v, Z s (y) = w. Theñ

Ricci-harmonic map heat flow in the continuous setting
We now assume, in addition to the assumptions (a),(b) and (ĉ) of the previous chapter, more regularity on d 0 andd 0 . Namely, we assume thatd 0 is generated by a continuous Riemannian metricg 0 on B 1 (0). This assumption will guarantee for all ε > 0 the existence of local maps defined on balls of radius r(ε), which are 1 + ε Bi-Lipschitz maps at t = 0. We explain this in the following Lemma.
Proof. The continuity ofg 0 means: for any ε > 0 and any x ∈ B 1 (0) we can find an r > 0 and a linear transformation, A : for all z, w ∈ B r/2 (A(x)). Returning to the original domain, we see that this means for all y, q ∈ B d0 (p, r/2) whereF 0 (p) = A(x), andF 0 = A • F 0 . This means in particular in view of the existence of the 1 + ε Bi-Lipschitz mapF , that (1 + c(n)ε) n ω n s n ≥ Vol(B d0 (p, s)) ≥ (1 − c(n)ε) n ω n s n for all s ≤ r.

Existence and estimates for the Ricci-DeTurck Flow with C 0 boundary data
In this section we construct solutions ℓ to the Dirichlet problem for the δ-Ricci-DeTurck flow on a Euclidean ball, which are smooth up to the boundary at time zero, and have C 0 parabolic boundary values. These solutions are constructed as a limit of smooth solutions ℓ α whose parabolic boundary values converge to those of ℓ.
Recall that the δ-Ricci-DeTurck flow equation for a smooth family of metrics ℓ is given by (see [9, p. 15] and/or [15, Lemma 2.1]) First we prove an estimate about the closeness of smooth solutions to δ in the C 0 norm, assuming C 0 closeness on the spatial boundary and a bound on the C 2 norm at time zero.
Proof. We will denote in the following | · | all norms induced by the metric δ. From smoothness and the boundary conditions, we know that ℓ is a smooth invertible metric for a small time interval [0, τ ] with |ℓ − δ| 2 ≤ ε(n) during this time interval. By (5.1) we can compute for all t ∈ [0, τ ] if ε(n) is sufficiently small. Thus by the maximum principle |ℓ−δ| 2 ≤ ε(n) remains true as long as this is true on the boundary. Thus we can take τ = T .
We now consider the problem of constructing solutions to the Dirichlet problem for the δ-Ricci-DeTurck flow, with boundary data h given on the parabolic boundary P of B R (0) × (0, T ). We will assume that the boundary data is given as the restriction of We now explain how to construct a solution to this Dirichlet problem if the compatibility conditions of the first type are satisfied. for all x ∈ ∂B R (0), where L l is the differential operator of order 2l which one obtains by differentiating (5.1) l-times with respect to t, and inserting iteratively the already obtained formulas for the m-th derivative in time for m = 1, . . . , l − 1. For example and We are now prepared to derive the following existence result. for all s ∈ (0, T ], x ∈ B R (0), for any given α ∈ (0, 1), where K : R + × R + → R is a monotone increasing function with respect to each of its argument.
We divide S by a small number δ(n) > 0, and call itS, i.e.
Hence, from the general theory of non-linear parabolic equations of second order, see for example [12,Theorem 7 Writing ℓ = h + δS and using the arguments above, we see that this will not be violated for t ∈ [0, T ], T ≤ 1, and that ℓ solves the δ-Ricci-DeTurck equation and ℓ| P = h| P .
This proves the existence of the solution. It remains to prove the Hölder boundary estimate, (5.2). For ease of reading, we assume R = 1.
If we assume that higher order compatibility conditions are satisfied, then we obtain more regularity of the solution.
Proof. The proof is the same, except at the step where we used [12, Theorem 7.1, Chapter VII] to obtain a solution in H 2+α,1+ α 2 , we now obtain a solution ℓ ∈ H k+α,k+ α 2 , in view of the fact that theS satisfies the compatibility condition of k-th order.
We now explain how to construct a δ-Ricci-DeTurck flow for parabolic boundary values given by h which do not necessarily satisfy compatibility conditions of the first order, but are smooth at t = 0, smooth on B R (0) × (0, T ] and continuous on B R (0) × [0, T ]. This is done by modifying the boundary values, so that the first (or higher order compatibility conditions) are satisfied, and then taking a limit.
Proof. Let ξ : R → R be a monotone non-increasing smooth function whose image is contained in [0, 1], such so that ξ is equal to 1 on [0, 1 2 ] and equal to 0 on [1, ∞).
6 An L 2 estimate for the Ricci-DeTurck flow, and applications thereof In this chapter we prove a lemma which estimates the change in the L 2 distance between two solutions to the δ-Ricci-DeTurck flow. Lemma 6.1 considers smooth solutions which are ε(n) close in the L 2 norm at time zero and agree at all times on the boundary. If we weigh the L 2 distance at time t of two smooth solutions appropriately, then this quantity is decreasing. The weight has the property that it is uniformly bounded between 1 and 2, and hence the unweighted L 2 distance at time t of the two solutions can only increase by a factor of at most 2. With the help of the L 2 -Lemma, we prove some uniqueness theorems for solutions to the δ-Ricci-DeTurck flow.
Remark 6.3. [7, Proposition 7.51] shows a uniqueness result of the Ricci-DeTurck flow with a background metric g with bounded curvature for solutions (g(t)) t∈(0,1) that behave as follows: there exists a positive constant A such that A −1 g ≤ g(t) ≤ Ag and |∇ g g(t)| + √ t|∇ g,2 g(t)| ≤ A for all t ∈ (0, 1). Its proof is based on the maximum principle. Corollary 6.2 assumes a stronger assumption on the closeness to the background Euclidean metric but it does not assume any a priori bounds on the first and second covariant derivatives : the proof is based on energy estimates.
By slightly modifying the previous proof, we can also show the following uniqueness statement.
We now make the further restriction, that the the metricg 0 is smooth on some Euclidean ball containing F 0 (x 0 ) in the sense of Definition 1.2. Theorem 1.3 shows in this case that the original Ricci flow solution comes out smoothly from some smooth initial data, if we restrict to a small enough neighbourhood of x 0 .
We return to the expanding gradient Ricci soliton examples provided by [14] and [8] discussed at the beginning of this section. By construction, they have non-negative Ricci curvature and bounded curvature at time t = 1 which amounts to saying that the corresponding Ricci flows satisfy (7.1).
We make a small digression to show that if an expanding gradient Ricci soliton satisfies (7.1) then it must have non-negative Ricci curvature. Indeed, let (M, g(t) = tϕ * t g) t∈(0,∞) be an expanding gradient Ricci soliton, satisfying 7.1 for all t ∈ (0, ∞). This clearly means that Ric(g(t)) ≥ 0: if this were not the case, say Ric(g)(x)(v, v) = −L < 0 for some x ∈ M and some vector v ∈ T x M of unit length with respect to g, then we must have Ric(g(t))(x t )(v t , v t ) = − L t g(t)(x t )(v t , v t ) for all t > 0 where x t := ϕ −1 t (x) and v t := (d xt ϕ t ) −1 (v). Consequently, Ric(g(t))(x t ) < −1 for t > 0 small enough, a contradiction. So without loss of generality, Ric(g(t)) ≥ 0 and hence the asymptotic volume ratio AVR(g(t)) := lim r→∞ Vol(B g(t) (x, r)) r n is well defined for all time t > 0 and all points x ∈ M by Bishop-Gromov's Theorem. Moreover, Hamilton, [7, Proposition 9.46], has shown that AVR(g(t)) is positive for all positive times t. Using the non-negativity of the Ricci curvature together with the soliton equation (7.2), one can show that the potential function is a proper strictly convex function. In particular, it admits a unique critical point p in M which is a global minimum. Since we are considering expanding gradient Ricci solitons, we know that (M, g(t), p) is isometric to (M, tg(1), p) as pointed metric spaces, and hence the asymptotic volume ratio AVR(g(t)) is a constant independent of time t > 0. Let (M, d 0 , o) be the well defined limit of (M, d(g(t)), p) as t → 0, the existence of which is explained in the introduction and guaranteed by [19,Lemma 3.1]. The theorem of Cheeger-Colding on volume convergence, now guarantees that the asymptotic volume ratio of (M, d 0 , o) is also AVR(g(1)) and that (M, d 0 , o) is a volume cone. In fact it is also a metric cone, due to [4,Theorem 7.6] and the fact that (M, d 0 , o) is the Gromov-Hausdorff limit of (M, td(g(1)), p) for any sequence t → 0. If x 0 ∈ M is a point where d 0 is locally smooth, in the sense explained in Definition 1.2, then (B g(0) (x 0 , r), g(t)) → (B g0 (x 0 , r), g 0 ) smoothly for some small r > 0 as t → 0.
In particular, if (M, d 0 , o) is a smooth cone, away from the tip o, in the sense that locally distance coordinates introduce a smooth structure near x 0 for any x 0 in M not in the tip of the cone, then the solution comes out smoothly from the cone away from the tip.