Ricci flow from some spaces with asymptotically cylindrical singularities
Abstract.
We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doublywarped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finitetime local singularities of Ricci flow, generalizing previous examples. The class also includes metrics with noncompact singular ends which become instantaneously compact. Furthermore, we prove local stability of the forward evolution, which allows us to glue it to other manifolds and create a forward evolution from spaces which are not globally warped products.
1. Introduction
For any complete Riemannnian manifold with bounded curvature, there is a smooth solution to the Ricci flow,
(1) 
with [Shi89]. The solution exists up to some time . In this paper, we prove the existence of a forward evolution of Ricci flow from a certain class of Riemannian manifolds with unbounded curvature. The initial metrics we consider have singular neighborhoods which are asymptotically cylindrical warped products of spheres.
Our primary motivation for considering this problem is the continuation of Ricci flow after singularities. The forward evolution of a smooth manifold often encounters local singularities in finite time. In some cases, the local singularities can be understood well enough so that Ricci flow with surgery can be implemented, e.g. [Ham97], [Per03], [Bre18]. In the three dimensional case the body of knowledge is by now quite powerful [KL14], [BK17].
All of these surgery examples work by proving that every local singularity encountered has a part close to a shrinking cylinder . The ideal situation for Ricci flow encountering such a singularity is when the metric is a warped product on for some interval :
(2) 
(Here is the standard metric on the factor.) By choosing correctly, the forward evolution from encounters a local singularity, named a neckpinch. This was conjectured in [Ham95] and first shown by Simon [Sim00]. In [AK04], [AK07] Angenent and Knopf expanded on these singularities and gave a precise asymptotic description. Their description in particular gives a description of the metric at the final time, when the singularity has occurred. In [ACK12], Angenent, Caputo, and Knopf proved the existence of a forward evolution of Ricci flow from these finaltime singular metrics.
The first main theorem in the present work provides the forward evolution from a family of singular metrics which includes those explored in [ACK12]. It also includes the forward evolution, in the ideal (doublywarped product) case, from (conjectured) singularities which are modeled on . Our description of the forward evolution is very precise, and we hope to provide a testbed for a more general theory that can deal with these singularities. We hope that our generalizations clarify the role played by various pieces.
The general question of which singular spaces have a forward Ricci flow has received attention from many authors. Particular success has been had with curvature bounds from below [Sim09], [Sim12], [CRW11], [Xu13], [BCRW17]. Another work addressing spaces with specific singularity models is [GS16]. For some results with low regularity on the initial metric, see [Sim02] and [KL12]. Furthermore, the Ricci flow of warped products lends itself to comparison to reactiondiffusion equations in Euclidean space, where there are quite general existence and uniqueness theories [GV97].
1.1. Model Pinches
We now give a definition of the singular metrics, which we call model pinches. Let , and let be the round sphere of sectional curvature 1, which satisfies for . Also let be any Einstein manifold with . The metrics will be metrics on of the form
(3) 
The main case of interest is but may be zero dimensional (landing us in the singly warped product case) or have negative Ricci curvature. The function will be increasing, so we can use as a coordinate and write
(4) 
Here .
For the rest of the paper we fix some . For any metric , function , and scale function we use the notation to mean the following. Take any point , scale the metric to , and then take the norm in the ball of radius around with respect to .
Definition 1.1.
A metric on of the form (4) is a model pinch if

[label=(MP0), ref=(MP0)]

As , .

If , there is a such that .

For some
(5) 
For any , on the set the curvature of is strictly bounded, and are and strictly positive, and is bounded.
One way to interpret this definition is that as the metric is asymptotically some sort of cylinder. At the distance scale given by , if the metric is close to the product , and if it is close to . For some precision, see Corollaries 2.8 and 2.9, which are stated for the forward evolution but in particular hold for the initial metric.
The Hölder condition implies the first and second derivatives of satisfy (for a different ) and similarly for . This allows for change by a factor of in the region where is .
Our first theorem is the following shorttime existence result in the class of warped products. We identify with and write .
Theorem 1.2.
The last sentence of the theorem gives a good description of the forward evolution near the origin, we will give an overview in Section 1.6. Here we just give a rapid tour of some properties that we think are important. In the forward evolution a small Bryant soliton appears at the origin, and the radius of the factor is strictly positive for positive time even if it was not for the initial metric . The distancesquared scale of the Bryant soliton is on the order of , and the largest Ricci curvature forward in time, which occurs at the origin, is on the order of . At the origin, the distancesquared scale of the factor is . If then this is at least order : property 2 says if , and if the term helps. This is not the case if and , so the largest Riemannian curvature could be on (and see cancellation in the Ricci curvature). Finally, we can use the control to provide a more precise rate of convergence of to ; see Corollary 4.13.
The next theorem removes the global warped product part of the model pinch assumption. For this, we need some additional assumptions on the curvature of the factor , which is inevitable since we allow perturbations of the initial metric in any direction. In particular, we rule out the case . For the metric , let . For example, if has dimension and constant sectional curvature then . In particular, .
Definition 1.3.
A model pinch is permissible if the following is satisfied.

[label=(RP0), ref=(RP0)]

In the case , we additionally require

In the case and (i.e. is flat) we additionally require as .
Theorem 1.4.
Let be an permissible model pinch. There is an depending on with the following property.
Let be a (possibly noncomplete) Riemannian manifold. Let be open, and assume that is a complete manifold with boundary, satisfying, for some and all , and .
Suppose that and is a diffeomorphism such that in ,
(6) 
Let be the differential manifold obtained by replacing with . For some , there is a Ricci flow , for on such that in as .
Immediate extensions of our theorems allow for multiple singular neighborhoods of each close to some model pinch, or for multiple extra warped factors each satisfying the requirements in the definition of model pinch.
1.2. Overview of the proofs
Both theorems are proven by constructing smooth mollified initial metrics, which agree with the singular initial metrics outside of a small set, and controlling the forward evolution of the smooth mollified metrics. By sending the size of the mollification to zero, we construct a forward evolution from the singular initial metric.
To prove Theorem 1.2, we control the relevant functions for the mollified initial metrics in terms of . The advantage of this is that the control is diffeomorphisminvariant– for example, the value of or at the point where is a diffeomorphisminvariant property. A usual difficulty in controlling solutions to Ricci flow is that the linearization is only weakly parabolic because of the diffeomorphism invariance of Ricci flow, and this gets around that issue. The most common response is to use RicciDeTurck flow, but we were not able to find a sufficiently good background metric to use in our case (and we tried some exotic possibilities). Another option in our case would be to use an arclength coordinate, but that introduces an annoying nonlocal term.
The forward evolution is split into two regions– the tip region, where a Bryant soliton forms, and the productish region, which includes the initial value and is where the metric continues to look locally like a product metric on . In Lemmas 2.2 and 3.2, we obtain local control in the productish and tip regions, assuming a priori boundary control. In Section 4 we put this control together. Section 4.1 shows that the boundary control needed at the right of the tip region is ensured by the local estimates in the productish region, and the boundary control needed at the left of the productish region is ensured by the local estimates in the tip region. We now have to prove the boundary conditions at the right boundary of the productish region, where the metrics are uniformly smooth. This is accomplished in Section 4.3.
The local estimates in the productish region use generic estimates for the solution to some reactiondiffusion equations in regions where they are nearly constant, which is dealt with in Appendix A. The local estimates in the tip region are more specific.
The proof of Theorem 1.4 uses RicciDeTurck flow around the alreadyconstructed warped product evolution to control an arbitrary metric. Theorem 6.1 is the main point in the proof, this gives us control of the RicciDeTurck flow in a neighborhood of the form , assuming a priori boundary control. Once we have Theorem 6.1, we wrap up by controlling the evolution in the boundary region (where everything is uniformly smooth) in Section 7.
Again, the local control in Theorem 6.1 is split into two parts: control in the productish region, and control in the tip region. As in the warpedproduct case, in the productish region we control the evolution using the results of Appendix A. On the other hand, in the tip region (where the solution is close to a small perturbation of the Bryant soliton), we use a contradictioncompactness argument to move the situation to the Bryant soliton. Then, we use a stability result for the Bryant soliton, Theorem 5.1. Theorem 5.1 might be compared to results from Section 7 of [BK17]; see the remark after the statement of the theorem.
1.3. Infinitely long pinched ends
In an attempt to simplify the initial exposition we have hidden that the left end of a model pinch, , may have infinite length. An arc length coordinate for the interval factor for a metric of the form (4) is given by , so the length is hidden in the integrability of near . In the case when the left end of the initial model pinch has infinite length, the left end of the evolution on is compact for positive time.
In two dimensions, Topping [Top11] constructed similar examples of noncompact surfaces which immediately become compact. These examples actually have initial metrics with bounded curvature, so it is especially interesting when compared with Shi’s existence result, which guarantees that the initial metric has a unique forward complete Ricci flow on the same topology. This means that in two dimensions there is an alternative, perhaps more natural, forward evolution besides the instantaneously compact one. In more than two dimensions, the analysis is different because the factor in the singly warped product has positive curvature and the initial metric must have unbounded curvature. We do not expect a natural forward evolution on the same topology in this case.
1.4. Some related shorttime existence results
Recent work that is close in spirit to ours is [Der16] and [GS16]. In [Der16], Deruelle showed that for any cone with positive curvature, i.e. a metric where , there is an expanding Ricci soliton which limits, backwards in time, to the cone. This can be considered as Ricci flow starting from the singular conical space. In [GS16], Gianniotis and Schulze allow us to start Ricci flow from any manifold which has local singularities modeled on these cones, by using local stabilitiy similarly to our Theorem 1.4. Such cones that are especially relevant to us are the singlywarped products , for ; these are singly warped products over intervals which are not covered by our theorem.
Alexakis, Chen, and Fournodavlos [ACF15] show the existence of a steady Ricci soliton of the form with . They also examine forward evolutions of metrics close to their steady Ricci soliton.
Bamler, CabezasRivas, and Wilking [BCRW17] examine the Ricci flow of manifolds with a variety of assumptions that curvature is bounded from below. In particular, they deal with complete, bounded curvature manifolds satisfying
(7) 
They show that there is a forward evolution for a time which only depends on and the dimension. An application is creating forward evolutions from singular spaces which can be approximated by manifolds with curvature bounded from below. This gives an alternative approach to some of the initial spaces considered by Gianniotis and Schulze in [GS16].
We wish to remark that we cannot apply the results in [BCRW17] in our case, but we need to use two different reasons. First note that in the examples with an infinitely long pinched end, the assumption on the volume of balls in (7) cannot be satisfied by approximating metrics, since the left end has balls of radius one with arbitrarily small volume. We claim that in the compact case the curvature condition in (7) is not satisfied. Consider just the singlywarped metrics of form (4), so . The curvature of such a metric is
(8) 
where and . The distance between and is
Note that the model pinches do (in the case when has positive curvature or ) satisfy an almostnonnegativity condition relevant to singularity analysis of Ricci flow, namely for a function satisfying as . This comes up, for example, in 12.1 of [Per02]. In this case is a multiple of and we can use the assumption 3 to bound , and the assumptions 2 and 3 to bound curvatures involving the other factor.
1.5. Model Pinches that arise as finaltime limits
Here we list some examples of smooth Ricci flows which have a model pinch has finaltime limits.
1.5.1. Singlywarped product singularities
In [AK07], Angenent and Knopf considered neckpinches occuring on singly warped products over an interval. They proved that the warping function of the finaltime limit of a neckpinch satisfies the asymptotics , where is the arclength from the singular end. This implies . Another singularity that may arise in the category of warped products of spheres over an interval is the degenerate neckpinch. In this case, Angenent, Isenberg, and Knopf showed in [AIK15] that the finaltime limit has the asymptotics where , . Forward evolutions from these specific cases were created in [ACK12] and [Car16], respectively.
1.5.2. Generalized cylinder singularities
For another example of a singularity, consider the doublywarped product depicted in the top row of Figure 2. A more stylized picture of a neighborhood of the singularity is Figure 3. The metric is a doubly warped product over an interval, with , and the singularity occurs at the left endpoint of the interval. Before the singular time, the metric satisfies the following boundary conditions at the left endpoint:
(9) 
Here is the distance from the left endpoint. A neighborhood of the left endpoint has topology before the singular time. For the initial metric, the size of the factor has a deep minimum at the center of the .
As time goes on, the factor shrinks drastically, and the metric encounters a singularity which can be rescaled to a generalized cylinder . Without rescaling, at the singular time the metric takes on the topology of the cone over (but is not asymptotically a metric cone). This singularity has not been rigorously constructed, but formal calculations suggest that the singular pinched metric should have asymptotics
(10) 
This is an unsurprising guess. The factor corresponding to the behaves similarly to a standard neckpinch. The dimensional part of the metric, , is close to being a flat , which corresponds to exactly. The flat metric is stable enough that the perturbation from the pinching factor does not affect it too much.
In the forward evolution of metrics with asymptotics (10), which we do investigate here, the size of the factor expands and the neighborhood takes on the topology .
1.5.3. Families of neckpinches
Here is another example which is a singularity modeled on , but which is qualitatively different from the previous. We can also consider a doublywarped product over an interval where has a neck somewhere in the interior of the interval. Then we can force a singularity to occur in the interior of the interval modeled on . Here there is an worth of onedimensional neckpinches forming. A trivial example of this is when we just cross a standard neckpinch with . While the previous example was also modeled on , this one is qualitatively different: for example, the topological change through the singularity is different.
This type of singularity should be stable in the class of doubly warped products; perturbations leave with a local minimum. However, in contrast to the previous example, it should not be stable in the full class of Riemannian metrics. It should not even be stable in the class of singly warped products where and is now an arbitrary metric on . (The original metric has .) Indeed, if we allow to also depend on the factor and perturb it so it has a strict local minimum at some point on that factor, we should approach a singularity at a single point on the factor. (Intuition for this may come from [MW85], which shows in particular that we can perturb the constant solution of simple reactiondiffusion equations on to get a single point blowup.)
1.5.4. Scarred neckpinches
Here is an example which leads to a metric which is not quite a model pinch. First consider a standard singly warped neckpinch with spheres of dimension : the initial metric is of the form and the metric at the singular time is a model pinch. This has a forward evolution, which recovers with a smooth disc of dimension at the tip. So, we have a Ricci flow of a singly warped product, at least on , for times , .
Now, the Ricci flow of warped products with Einstein fibers does not care about the Riemannian curvature tensor of the fiber metric, it only cares about the Ricci curvature. In other words: suppose we have a Ricci flow on of the form (where for each , ) and . Suppose is another Einstein manifold with . Then is also a Ricci flow.
Therefore, in the Ricci flow through a standard neckpinch, we can swap out with any Einstein manifold of our choosing, provided it has the same scalar curvature as . The resulting object satisfies Ricci flow wherever , but is not a manifold for . Around the new points at the tip, the result has the topology of the cone over . The forward evolution has a scar as a result of its surgery.
A special case of this situation is when is the standard metric on for some group . This case is important because it cannot be ruled out by a pointwise curvature condition, and so it is relevant to trying to implement Ricci flow with surgery under curvature assumptions. The resulting object after the singularity is an orbifold. This case was dealt with in four dimensions in [CTZ12], and they removed a topological assumption of Hamilton’s work in [Ham97] by considering Ricci flow with orbifold singularities.
Of relevance to us is the case and . In this case, the metric at the singular time has the form ^{1}^{1}1 We are always lazy with writing the lifts of metrics and tensors etc. Here we use the notation to emphasize that the two terms which appear are different, one is the lift of the on the first factor, and the other is the list of the on the second factor.
(11) 
It satisfies all of the conditions of a model pinch except for 2, since and . Since is unstable under Ricci flow (we can perturb the size of one of the factors) we thought perhaps there could be two alternative forward evolutions where either of the factors becomes positive after the singular time. We now believe that this is not possible, see Section 1.7.3.
1.6. Shape of the forward evolution
In this section we describe various properties of the forward evolution of a model pinch. As time goes on, the metric continues to be a doubly warped product:
(12) 
Furthermore, we prove that continues to be increasing in . Therefore we may continue to consider and as functions of , and write the metric as
(13) 
For the initial metric, the derivatives of and are relatively small. Therefore after investigating the curvature of warped products we see
(14) 
Forward in time, this approximation continues to hold for a short time, while the derivatives of and continue to be small. We call the region where continues to be small the “productish” region. Let . The productish region is the set
(15) 
for some sufficiently large and small . In this region, we have ; by choosing and we can have as small as we wish.
In the productish region, we get the approximations
(16)  
(17) 
Note that these approximations would be exact if the approximation (14) were exact so , , and
(18) 
In Section 2.4 we give some corollaries of our control in the productish region.
Now we come to a crucial juncture in the calculation of our approximate solution. The approximations (16) and (17) work for  in particular they work for . To understand the approximations for small , put , and write
(19)  
(20) 
Using our assumptions on and , particularly 3, we can estimate the quotients for . Then our approximations say
(21)  
(22) 
where , , .
If the left end of the manifold is to be smooth and compact, cannot be small up to . In fact, is a necessary condition to have a smooth closed disc at the left endpoint. At the left end, on the factor , we glue in a steady Bryant soliton of size . This is a metric on that moves only by diffeomorphisms under Ricci flow. We call the region where stays small, where we see the Bryant soliton, the “tip region”. The asymptotics of the Bryant soliton as match with the term in (21). A steady soliton is in accordance with the fact that we expect scaling at a rate faster than : as a general principle, if we scaled at rate we would expect an expanding soliton, whereas if we scale at a faster rate we find a steady soliton.
For the factor , the warping function is approximately constant. Therefore we expect to be able to attach a large factor to our Bryant soliton. The approximate size of the unrescaled factor is . Taking for simplicity the case , our assumptions imply that . Therefore when we scale by the size of this factor goes to infinity, and around any point it approaches a Euclidean factor.
Thus, the zeroth order approximation of the metric near the tip (in other words, the expected limit of the rescaled metric as ) is . We can get this approximation in a region of the form
(23) 
As (so ) this region covers the whole Bryant soliton.
We also need to find the first order approximation near the tip. The perturbation has size . The equation we get in space is
(24) 
where and represent the zeroth and first order approximations. This gives us an equation to solve for . On the factor, the solution coincides with the soliton potential, times . Our first order approximation matches with all of the terms in (21), (22).
In Section 3.9 we give some corollaries of our control in the tip region.
1.7. Sharpness and further questions
1.7.1. Regularity conditions 3
Note that an implication of is that can be bounded for small , independently of . In particular, and both do not satisfy our assumptions. We cannot offer any guess as to whether our results hold for these functions.
As examples of wild profiles for , consider or . Note that if the initial metric has bounded length near , then this may appear bad. Still, around any point where , rescaling by we will see approximately a product metric on a long (length ) scale. Note in all cases, in the forward evolution is bounded and positive near the tip for finite time.
We can think of the conditions on in the same way, but it may be more reasonable to look at examples in terms of the arclength coordinate . So, consider the part of the metric written as
(25) 
The condition that actually says, in a sense, that must be small enough in terms of . (Written in terms of , this condition will involve the functional inverse of .) The following functions satisfy the regularity conditions on :

and , where and , or and .

and , where and .

If we write where as , then the condition that is equivalent to . For example, or , are both valid model pinches.
1.7.2. The profile
Our results do not provide a forward evolution from the initial metric with , and at . Note in that case
(26) 
so . Then , which violates condition 3. It would be interesting to know whether there is a solution to Ricci flow emerging from this example.
In this example, for any , the region which looks approximately like a skinny cylinder of radius is quite long in comparison to . More precisely, fixing there is a such that for any we have the following. The region where the radius is within a factor of of has length . Maybe this means the cylinder must collapse before anything far away can save it.
1.7.3. The conditions on the size of 2
For simplicity say . We find it striking that in the case , for the initial metric and are comparable, but if we rescale the forward evolution to keep the curvature bounded at the origin, the factor goes to infinity.
We believe that it is possible to relax the condition 2 and still have a forward evolution with the same asymptotics. Let’s rapidly go through a calculation. Suppose , where (violating 2). Calculating from (17), in the productish region where ,
(27)  
(28) 
If we write then for points where we have (recall ):
(29) 
First consider the case (i.e. ), which is still a weaker condition than 2. Then scaling in the same way we scale sends it to infinity, and is approximately a constant. We expect this case to behave similarly to the case that is rigorously dealt with in this paper. The major road block in dealing with it, for us, is reproving Lemma 3.5 which controls the derivative of and therefore controls the level of interaction between the evolution of and . Unfortunately our method gives us no more wiggle room in this lemma, but we think that our control on the distance from to a constant is not optimal.
To continue with our speculation, consider the case when . Then in (29) we find . We still would have the approximation (21) for . This gives us the asymptotics for an Ivey soliton [Ive94], which is a complete soliton on of the form . (The function goes to zero at , and stays positive.) So, in this case we expect to see the Ivey soliton in the rescaled limit at the tip. This case should be more difficult, because the system is more strongly coupled.
In the case when , we do not think that there is a smooth forward evolution, but there may be a forward evolution with bounded Ricci curvature everywhere. In this forward evolution we glue in a Bryant of dimension , but with the sphere fibers replaced with the Einstein manifold (with proper scaling to make the scalar curvatures match). The case is the situation discussed in Section 1.5.4.
The reason we do not expect a smooth forward evolution is the following: consider . Then, we are in the case when we expect the Ivey soliton. The exact asymptotics of the Ivey soliton we get are determined by , and as , this family of Ivey solitons approaches the Bryant soliton with replaced with . Therefore, even trying to approximate the singular initial metric with smooth ones it seems we are led to the nonsmooth case.
1.7.4. Pinched sphere warped products over other bases
Consider a manifold with boundary with a metric and a function which tends to zero at the boundary such that also goes to zero. Let’s stipulate that everywhere . Now we want to ask whether there is a forward evolution from the metric . Note that model pinches with are a special case, where .
The nice property of the doubly warped products is that the hessian of is easier to control, because the level sets of are equidistant. It should be relatively possible to extend to other such cases, like cohomogeneityone manifolds.
1.7.5. The closeness required in the asymmetric case
Our condition for Theorem 1.4 is that the distance between the asymmetric metric and the model pinch goes to zero near the tip at least as fast as a specific rate. There is a sense in which this is probably not optimal. Our proof technique yields more than is stated in Theorem 1.4: it says that actually stays close to the forward evolution from . We make no attempt to update the approximate model pinch, whereas perhaps the best warpedproduct forward evolution not the forward evolution from the initial warpedproduct.
A theorem that we can compare Theorem 1.4 to is Theorem 1.3 of [GS16]. That theorem constructs forward evolution from metrics close to having conical singularities. There, is a cone and the requirement (1.1) is that near the singularity the singular metric satisfies . This seems stronger than our theorem, because it makes no exact assumption on the rate at which it approaches the model singularity. On the other hand, the case of a singlywarped cone (which our theorem does not handle) is the case when is constant, so perhaps our condition is not dissimilar.
1.8. Notation and preliminaries
More notation is densely listed in Appendix E.
Partial derivatives are denoted with . For an arbitrary function with nonzero derivative, we have which is the derivative with respect to , using a metric. We define which is the derivative with respect to time along a curve which moves orthogonally to the level sets of in order to keep constant.
We adopt the shorthand that when stating hypotheses, the statement “” means “there exists an , depending on and , such that if , the following holds.” This allows us to quickly state “if and then …”.
1.8.1. Equations
We can consider our metrics as singly warped products of spheres over a general base: where for each , . Under Ricci flow, evolves by
(30) 
where is the heat operator and is the laplacian for . Equivalently,
(31) 
where is the heat operator for . Similarly, the function which controls the size of evolves by
(32) 
where we have defined . We use this point of view to find the approximate solutions in the productish region. For an exposition of these equations for Ricci flow on warped products, see Section C.
1.8.2. Regularity
We work in Hölder spaces using interior Schauder estimates. Bamler wrote a clean statement of the interior Schauder estimates he needed in [Bam14] (Section 2.5). We coopt this statement, because it is exactly what we need except for standard generalizations. His statement does not allow for the timedependence of the coefficients that we will have, but in fact the proof carries through exactly; the time dependence enters in the estimate on the norm of in the middle of page 424. Furthermore, his statement does not allow the parabolic ball to hit the initial time, as we will need to. Accounting for this is also standard. In the proof of Lemma 2.6 of [Bam14], one may apply Exercise 9.2.5 of [Kry91] rather than Theorem 8.11.1 of [Kry91].
1.8.3. RicciDeTurck flow
We use RicciDeTurck flow to control the Ricci flow of metrics near our warped product forward evolutions. For two metrics and we define
(37) 
which is the map Laplacian of the identity map from to . For a timedependent metric we define . The RicciDeTurck flow from with background metric is the solution to
(38)  
(39) 
We allow to also be timedependent. It will be useful to consider Ricci flow and RicciDeTurck flow modified by a vector field. We set , and if then we say that is a solution to RicciDeTurck flow, modified by , with background metric .
We will not use the exact form of the evolution of , except to know that we can apply regularity. What we will use is the following evolution of and . For We set
(40) 
Now, assuming that , and that , for we have (we allow to change from line to line)
(41) 
For we have
(42) 
We show these in Appendix D.
2. Control in the productish region
In this section we create some interior estimates for our warpedproduct forward evolution. We define the productish region as a region of the form
(43) 
In particular, touches an open part of the initial time slice (see Figure 1). All constants and definitions in this section implicitly depend on dimensions, , and the chosen functions satisfying the model pinch conditions and . We define . In the productish region, we will have approximations of the form
(44) 
These come directly from the calculations in Appendix A. They may be guessed by ignoring all terms in the evolution of and which depend on space derivatives of or . We will prove that is between and , and is between and , where
(45) 
We call and the barriers.
We make some definitions to state the main result of this section. We will assume that is a solution to Ricci flow on . Our definitions depend on constants , , controlling the size of the produtish region, and controlling the separation of the barriers, as well as and .
Definition 2.1.
We say that is barricaded (by the productish barriers) ^{2}^{2}2In this section we only say “barricaded” but in Section 4 we will have to refer to either barricaded by the productish barriers, or barricaded by the tip barriers. at a point if it satisfies and at that point.
We say that is initially controlled in the productish region if at and for all points satisfying it is barricaded and