The Ricci flow on manifolds with boundary
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.
The aim of this paper is to study the deformation for a short period of time of a Riemannian metric on a compact Riemannian manifold with boundary using the Ricci flow
which was introduced by Hamilton in . He established the short-time existence and uniqueness of solutions with and used it to study three-dimensional manifolds admitting metrics with positive Ricci curvature. Later on, Shi in  proved the short time existence of the flow for complete manifolds with uniformly bounded Riemann tensor. Ever since, it has been proven to be a valuable tool in the study of the interaction between geometry and topology, providing a natural geometric deformation of Riemannian manifolds.
A natural question to ask is whether one can deform the geometry of a manifold with boundary using the Ricci flow, and what would be appropriate boundary conditions. The obstacle, as in the case without boundary, is the diffeomorphism invariance of the Ricci tensor which is why the equation is not parabolic. One needs to solve a modified parabolic equation first, as DeTurck did in  and then relate its solution to the Ricci flow. In the case of manifolds with boundary though, the challenge is to impose boundary conditions that on the one hand will lead to a parabolic boundary value problem for the modified equation, and at the same time tie well with the geometric character of the Ricci flow.
The first work in this direction was by Y.Shen in , where he established a short-time existence result for compact manifolds with umbilic boundary. Moreover, he extended Hamilton’s result in  to the case of manifolds with totally geodesic boundary. The convex (and umbilic) case was studied later by Cortissoz in . However, one would like to deform more arbitrary metrics than in . To this direction, Pulemotov in  proved a short-time existence result for manifolds with boundary of merely constant mean curvature.
More work has been done on the two-dimensional Ricci flow, and the closely related Yamabe flow. Both have been studied under Neumann-type boundary conditions. See for instance the contributions of Brendle in , , , Tong Li in  and Cortissoz in . Also, Giesen and Topping in  study the Ricci flow on general incomplete surfaces from a different point of view. They show existence of solutions which become instantaneously complete for positive time and completely classify their asymptotic behaviour. Moreover, Topping in  shows that such flows depend uniquely on the initial data.
Heuristically, the Ricci flow is closely related to the corresponding “elliptic” problem, the Einstein equations. Boundary value problems for Einstein metrics have been studied for instance by Anderson in , Anderson and Khuri in , Schlenker in  and Reula in . In particular, in  it is shown that the conformal class and the mean curvature of the boundary give elliptic boundary conditions for the Bianchi-gauged Einstein equation. Notice that in the case of three-dimensional manifolds with boundary, solving such a boundary value problem also gives rise to immersions of the boundary data (conformal class and mean curvature) in the canonical simply connected spaces of constant curvature. We refer the reader to  for details on this point of view. A parabolic approach may provide further understanding of these geometric problems.
A solution to the Ricci flow is not expected to be determined uniquely by the mean curvature only, as in , which hints that it should be supplemented with additional boundary data. In the following, we study boundary value problems for the Ricci-DeTurck flow and the Ricci flow, under the boundary conditions proposed in . The main result of this study is a local existence and uniqueness result for the Ricci flow on manifolds with boundary. To the knowledge of the author, this is the first result which allows the flow to start from an arbitrary initial metric.
The methods used can also be applied to study boundary value problems for geometric flows related to static metrics in General Relativity (see ). However, we won’t pursue this direction here, as we plan to discuss it in a future paper.
Let be a compact dimensional manifold with boundary and interior . If is a smooth Riemannian metric on we will denote by the mean curvature of the boundary and by the part of the tensor , tangential to the boundary. Moreover, if is some Riemannian metric on , let be its conformal class, namely
Now, let be an arbitrary smooth Riemannian metric on , a smooth time-dependent family of metrics on and a function . We assume that they satisfy the zeroth order compatibility conditions
Moreover, let be the constant defined in Section 4.1, which bounds the norm of and appropriate Hölder norms of and . For the precise definitions of the function spaces used and the restrictions on the values of and appearing below the reader is invited to consult Section 2.
Let be as above. Consider an arbitrary family of background metrics which satisfies in addition the zeroth order compatibility condition . Take and set
Then, there exists a which depends only on and and a unique solution , , of the Ricci-DeTurck equation,
where , satisfying on the boundary conditions:
and the estimate . The solution is away from the corner , and extends on as a family of metrics. Moreover, if the data , , and satisfy the necessary higher order compatibility conditions (see Section 4.2), then is up to .
Now, Theorem 1.1 allows us to prove in Section 5 a short-time existence result for the Ricci flow on an arbitrary compact Riemannian manifold with boundary. Here, the existence time of the flow is controlled in terms of bounds on the geometry of the initial data.
Let , , as in Theorem 1.1, and suppose
Moreover, as , converges in the Cheeger-Gromov sense to and away from the boundary. Namely, there exist a smooth family of diffeomorphisms of , , such that .
Also, if satisfy the necessary higher order compatibility conditions for the Ricci tensor to be in the class (see Section 5), then
As , converges to in the Cheeger-Gromov sense.
, and there exists a diffeomorphism of which fixes the boundary and is in the interior such that . Also, if , is and .
The Riemann tensor is in and .
Here, denote the injectivity radii of , respectively and denotes the “boundary injectivity radius”, namely the maximal size of the collar neighbourhood of in which the normal exponential map from the boundary is a local diffeomorphism. See Definition 5.1. Also, we write for the scalar curvature of .
We note that a version of Theorem 1.2 in which the initial data are obtained in the usual sense, namely , does hold. However, such a solution will generally not be smooth up to the boundary even for positive time. This issue is related to the invariance of the equation under diffeomorphisms and is discussed in Remark 5.3.
We prove Theorem 1.1 in Section 4 with a fixed-point argument, following the method of Weidemaier in  and applying Solonnikov’s work on linear parabolic systems under general boundary conditions in . The main advantage compared to an implicit function theorem approach is that the study of the nonlinearities of the equation and the boundary conditions allows us to obtain uniform control on the existence time.
Note that the control of the existence time obtained in Theorem 1.1 does not tie well with the geometric nature of the Ricci flow, mainly because it involves norms which depend on the choice of the background smooth structure and metric. From this point of view, Theorem 1.2 is more satisfactory, as the lower bound on the existence time depends only on the geometry of the initial data and norms of the boundary conditions.
It is well known that incomplete solutions of the Ricci flow are in general not unique. On a manifold with boundary though, the boundary data (1.5)-(1.6) allow us to obtain the following uniqueness result.
Finally, in Sections 6 and 7 we move towards the study of more global issues. In Section 6 we demonstrate the necessity of the bound on the boundary injectivity radius in Theorem 1.2. We construct examples with flat initial data and uniformly controlled boundary conditions whose existence time becomes arbitrarily small. This is quite surprising, since on closed manifolds a curvature bound suffices to prevent such behaviour. Section 7 is devoted in the proof of the following theorem, which is a continuation principle for the Ricci flow on manifolds with boundary.
Let , be a smooth () Ricci flow on with smooth boundary data defined for . Suppose be the maximal time of existence and . Then
Acknowledgements: The author would like to thank his adviser Michael Anderson for suggesting this problem and for valuable discussions and comments.
2. Notation, definitions, background material
Let be a smooth, compact, dimensional manifold with boundary , and interior . We will use the notation , .
2.1. Function Spaces.
We need to define the function spaces we will use. First, fix a smooth Riemannian metric on and denote by its Levi-Civita connection. We also need to fix an open cover of , and a collection of charts such that
|, if does not intersect the boundary|
|, if intersects the boundary.|
In the last case assume that . We will use the convention that Greek indices correspond to directions tangent to the boundary, counting from to . Moreover, will be a partition of unity subordinate to that open cover.
Consider any tensor bundle of rank over , with projection map , equipped with the connection inherited by . The completion of the space of the time dependent sections of with respect to the norm
will be denoted by . Let also
If is a section of , we will denote by the coordinates of this tensor with respect to the trivialization based at .
We define the following norm for time dependent sections of and for :
Here, setting , we define
where, for every function
In the above,
Analogous spaces exist also in the elliptic setting, see for instance .
For nonintegral, we will denote by the Banach space of time dependent sections of having continuous up to the boundary derivatives for all satisfying , satisfying appropriate Hölder conditions in the time and space directions. More precisely, the norm is given by
where are the coordinate functions of in the coordinate system and
We will also denote by and the norms
For any integer we will denote by the space of sections with all the derivatives for continuous, equipped with the norm .
By the definition of it is not hard to see that embeds in , provided that . We will also need the following embedding theorems.
For , and ,
If and , then
If and , then
In the above, the constants do not depend on and , where is a constant depending on the chosen atlas .
From now on let us fix some and some . Then, as the previous Lemma implies, the Sobolev space embeds in the Hölder space . Moreover, we get the following estimates (see Corollary A.2 in ).
For all , with , , and all sufficiently small
, for all
Also, we will be using the following product estimate.
If and , then
2.2. The mean curvature.
Let be a Riemannian metric on and the ourward unit normal to with respect to . The second fundamental form of the boundary is defined by
The mean curvature of the boundary with respect to the metric is then given by
In the following we are going to need the following formulae, which can be proven by direct computation.
If is a smooth one-parameter family of metrics, such that , and , the first variation of the mean curvature of the boundary is given by the formula:
In the local coordinates defined in this section the mean curvature of the boundary of is given by
3. A linear parabolic initial-boundary value problem.
Let be a Riemannian metric on , for some , the induced metric on the boundary, the Bianchi operator and be the linearization of the mean curvature at .
We will also denote by and the subspaces whose elements satisfy the initial condition .
Consider the following linear parabolic initial-boundary value problem on symmetric 2-tensors on
for , in the corresponding space and . Assuming that the zeroth order compatibility conditions
hold, problem (3.1) has a unique solution which satisfies the estimate
Moreover the constant stays bounded as and depends on the norms of and .
The method followed in Chapter IV of  and Theorem 5.4 of  carries over to the manifold setting, after the necessary adaptation to the realm of manifolds and vector bundles (see ). We only need to show that the following boundary value problem on satisfies the complementing condition (see , and ).
Here, and . One obtains (3) by expressing (3.1) in local coordinates around a point of the boundary, with , freezing the coefficients at and keeping the higher order terms. The principal symbols of the boundary operators are:
and the principal symbol of the parabolic operator , is , where and its Euclidean norm. We obtain the following positive root . Setting equations to zero and letting , , we get the following system:
for some .
From equation we have
while multiplying equation by and then adding over we find:
This gives, taking (3.10) and into account:
which, after substituting for , leads to the equation:
Now, by equation we have:
which combined with gives:
and therefore . Now, (3.9) implies that , which gives and thus .
Now, by (3.13) we have that
However, assumption (3.9) implies that , since for . This gives that .
Theorem 3.1 is still valid if we consider and evolving such that . Note that the complementing condition is satisfied if and are in the same conformal class. If not, the openness of this condition implies that it holds at least for some short time depending on bounds of and . Thus, we either get local (in time) existence or a global solution and the constant depends on the norms of and and the norms of and .
4. A boundary value problem for the Ricci-DeTurck flow.
Let be a Riemannian metric on . Consider also , a family of boundary metrics and a function , where is always and . Moreover, assume the zeroth order compatibility conditions (1.2) hold.
We supplement the Ricci flow equation
with the boundary conditions
and the initial condition
and aim to study the existence and regularity of solutions.
As is well known, the Ricci flow equation is not strongly parabolic, so we will first study the Ricci-DeTurck equation
with the boundary conditions
Here, , being the Christoffel symbols of , and the Christoffel symbols of a family of metrics with (i.e ).
The geometric nature of Ricci flow requires the boundary data to be geometric, namely invariant under diffeomorphisms that fix the boundary. The data (4.2) have this property. However, passing to the DeTurck equation we need to impose the additional, gauge-dependent boundary condition .
We allow the background metric to vary and define a time dependent reference gauge. This, as will be discussed in Section 4.2, allows higher regularity of the solution on .
4.1. Short-time existence of the Ricci-DeTurck flow.
We can now state and prove the main short time existence Theorem. First, define
Then, the following theorem holds.
Consider the boundary value problem , with initial condition . For the data define
For any there exists a and a solution of this initial-boundary value problem which satisfies .