In a previous post about the proof of the differentiable sphere theorem (due to S. Brendle and R. Schoen), we saw that a fundamental ingredient was the recent technique of deformation of Ricci flow invariant cones of ”curvature conditions” introduced by C. Bohm and B. Wilking. Today we will discuss a little bit more of the details of this interesting new method of Bohm and Wilking for the analysis of the Ricci flow.
Let me start with some historical remarks: Ricci flow was introduced in 1982 by R. Hamilton in order to show that compact 3-manifolds with positive Ricci curvature admit metrics of positive constant sectional curvature (i.e., they are spherical space-forms). See a link to Hamilton’s article here. This gave a positive answer to a question of J. Bourguignon). Also, in 1986, Hamilton obtained similar conclusions in dimension 4 for positive curvature operators. Nevertheless, H. Chen generalized Hamilton’s result by showing that compact 4-manifolds with 2-positive curvature operator are spherical space-forms (recall that 2-positive curvature means that the sum of the two smallest eigenvalues of the curvature operator is positive). Finally, in view of these results, Hamilton conjectured that compact manifolds of any dimension with positive curvature operators are spherical space-forms.
In this direction, Bohm and Wilking introduced a new important tool allowing to confirm Hamilton’s conjecture. More precisely, they showed the following theorem:
During the rest of this post, we will discuss some aspects of Bohm and Wilking’s technique and its application to the proof of the theorem above. To do so, we divide the post into four sections: in the next section we will outline Bohm and Wilking’s argument; after that, we dedicate the subsequent section to review relevant facts concerning the algebraic properties of curvature operators; the third section will contain a description of Bohm and Wilking construction of ”pinching families” of cones of curvature operators and the last section contains a brief exposition of the application of this pinching families method to the proof of the desired theorem.
–Outline of Bohm and Wilking method–
Recall that the Ricci flow is the evolution equation
where is a curve of Riemannian metrics on a manifold . Following R. Hamilton (see also this book of B. Chow and D. Knopf), one can use moving frames (a.k.a. Uhlenbeck’s trick) to obtain that the curvature operators of the metrics satisfies the following evolution equation:
where (here we are using the identification and the adjoint representation).
Remark 1. In the sequel, we normalize the constants so that the curvature operator of the round sphere of radius 1 is the identity.
A important tool for the study of the Ricci flow is the so-called ”Hamilton’s maximum principle”: it says that a closed convex -invariant subset of curvature operators staying invariant under the ODE
defines a Ricci flow invariant curvature condition. In other words, if the curvature operator of the initial metric belongs to a subset which remains invariant under (1), then the curvature operator of every also belongs to (i.e., curvature constraints of preserved by (1) are not destroyed by the Ricci flow). Of course, it is not easy to apply Hamilton’s maximum principle in higher dimensions because the ODE (1) is not well-understood (this explains why the results of Hamilton and Chen quoted in the beginning are restricted to dimensions 3 and 4).
Nevertheless, Bohm and Wilking were able to exploit Hamilton’s maximum principle in the following way: we start with given by the closure of the open cone of 2-positive curvature operators ( is taken closed just to fit the assumptions of Hamilton’s maximum principle) and we would like to show that any metric whose curvature operator belongs to the interior of evolves by the Ricci flow to a metric of positive constant sectional curvature. Assume that one can construct a family of invariant cones , , starting at , such that the vector field always points towards the interior of for any and converges to when (with being the identity curvature operator).
Remark 2. At this point I strongly recommend the reader to make some qualitative pictures of an ODE associated to a vector field with these properties (since I don’t know how to insert xfig files in WordPress posts 😦 ).
As the reader can from the pictures and Hamilton’s maximum principle, the existence of such a family of cones of curvature operators guarantees (intuitively) that the flow generated by the ODE (1) evolves any metric inside to a metric with curvature operator equals to a constant multiple of the identity, i.e., a metric of constant sectional curvature. Another way of thinking about this argument passes by a continuity argument: we begin with the cone and we run the Ricci flow. Since the vector field points towards the interior of , we evolve to a curvature operator inside a cone for some . Moreover, from Hamilton’s maximum principle, each time we attain a given cone , our future evolution is faded to stay inside the cone forever. Nevertheless, we know that the vector field points towards the interior of , so that we eventually reach a smaller cone for some . Reproducing this argument ad infinitum (and crossing fingers 🙂 ) we ”conclude” that the Ricci flow will send our initial metric (whose operator belongs to ) to a metric inside the limit cone , so that the proof would be complete.
Thus, it remains ”only” to justify (heuristically, at least) why one should expect such a family of cones to exist. A (very) naive idea consists into directly connect a nonnegative curvature operator to a multiple of identity part, say , where , is the Ricci tensor of , and consider the family of cones . Observe that converges to when (since when ). However, it is not obvious that this family is invariant by the ODE (1) and the vector field is always pointing inward. In order to check these two conditions, one can take the pull-back of the vector-field by . It follows that is invariant by the ODE (1) if and only if belongs to the tangent cone of at . On the other hand, since we are assuming that , it suffices to get that the difference
lies inside . It turns out that this difference can be explicitly computed. Moreover, this difference depends only on the Ricci tensor of (but not on its Weyl tensor) making the computations quite pleasant. However, a quick inspection of the formulas shows that this family is not invariant by the ODE (1) essentially because this difference has (very) negative scalar curvature for large (so that it can’t stay inside the cone of 2-positive curvature operators). To overcome this technical difficulty, Bohm and Wilking introduce a more general family of linear operators
where is the traceless part of the Ricci tensor of . Here we use and instead of and just for normalization purposes. Again, we look at the associated family of cones . It is not hard to see that, for each fixed , the family converges to . Taking the pull-back of the vector field under and looking at the corresponding cone , we see again that this family of cones is invariant by the ODE (1) if and only if the difference
lies on . It turns out that can be computed explicitly (essentially because it depends only on the Ricci tensor) and we can adjust the values of , so that belongs to (i.e., this family of cones is invariant by (1)) and join to (this is a specially tricky part of the proof, but it is not terrible difficult once one get an explicit formula for ). In particular, this allows to get the desired family of cones, so that our sketch of the Bohm and Wilking arguments is complete.
Now let us turn to the details.
Denote by the vector-space of curvature operators, i.e., symmetric endomorphisms of verifying the first Bianchi identity (so that in our notation , stands for ”symmetric” and stands for ”Bianchi”). Consider the linear transformation introduced in the previous section:
and the difference
between the pull-back and . As we discussed above, one relevant step in Bohm and Wilking argument consists into deriving an explicit formula for :
Theorem 1 (theorem 2 of Bohm and Wilking). .
Remark 3. Putting and , we see that the difference is exactly . Using this theorem, we see that the scalar curvature of (i.e., the quantity in front of the operator I) becomes negative when , so that Hamilton’s maximum principle can’t be applied to the related family of cones (in particular this family is not invariant by the ODE (1), as we announced in the previous section).
The proof of this theorem is not hard but it involves a certain amount of calculation. Thus, we will just briefly indicate the main steps.
Proof of theorem 1 (sketch). One begin by calculating the eigenvalues of the linearization of the vector-field around the identity . It is not hard to see that
See lemma 2.1 of Bohm and Wilking. After that, one applies this formula for an explicit calculation of the vector-field itself when the curvature operator is a Ricci operator i.e. . The resulting formula is
where and . See lemma 2.2 of Bohm and Wilking. Finally, combining some linear algebra arguments (based on the canonical decomposition of a curvature operator into a multiple of the identity , its traceless Ricci tensor and its Weyl tensor ) with this formula for (when is Ricci), one deduces the desired expression for the difference . See the proof of theorem 2 of Bohm and Wilking. This completes our sketch of the proof of theorem 1.
An interesting corollary of this theorem is:
Corollary 1 (corollary 2.3 of Bohm and Wilking). Let be an orthonormal basis of eigenvectors associated to the eigenvalues of . Then, is an eigenvector of associated to the eigenvalue
where and . Also, is an eigenvector of the Ricci tensor of associated to the eigenvalue
The relevance of this corollary is that the knowledge of the eigenvalues of will guide our choices of parameters during the construction of the desired family of cones.
–Construction of pinching families of cones of curvature operators–
Before starting the discussion of this section, let us recall (in a more precise fashion) the properties of the desired family of cones:
Definition 1. A continuous family of closed convex -invariant cones (of full dimension) is called a pinching family whenever
- each has positive scalar curvature,
- stays inside the interior of the tangent cone of at for any and ,
- converges (in the Hausdorff topology) to the 1-dimensional cone when .
Example 1. The family of cones of 3-dimensional curvature operators
is a pinching family starting at the cone of curvature operators (in dimension 3) with nonnegative Ricci curvature. This family of cones was used by R. Hamilton in order to show that the Ricci flow on 3-manifolds evolves any metric with positive Ricci curvature to a limit metric with positive constant sectional curvature.
Roughly speaking, the goal of this section is to show how one can use theorem 1 (and corollary 1) to generalize the previous example to the higher dimensional case.
Theorem 2 (theorem 3.1 of Bohm and Wilking). There exists a pinching family , of closed convex cones starting at the cone of 2-nonnegative curvature operators.
This pinching family will be defined piecewise by 3 subfamilies. For the construction of the first subfamily, we need the following proposition:
Proposition 1 (proposition 3.2 of Bohm and Wilking). Take and let be a closed convex -invariant cone of curvature operators which is invariant by the ODE (1). Assume that contains all nonnegative curvature operators of rank 1, any has nonnegative Ricci curvature and is contained in the half-space of positive scalar curvature operators. Then, the cone is invariant by the vector-field associated to the ODE (1) for
Proof. It suffices to show that belongs to the interior of the tangent cone . Since (because is invariant by the ODE), our task is reduce to prove that . Since contains all nonnegative curvature operators of rank 1 (and consequently its linear combinations because is convex), we are done if one can show that is positive for . Looking at the eigenvalues of computed in corollary 1 and using the definition of in terms of , we see that is positive if and only if , which holds in the range as the reader can directly check.
Since it is known that the cone of 2-positive curvature operators is invariant by the ODE (1) (see the paper of Hamilton), we can apply the previous proposition to get the following consequence:
Lemma 1 (corollary 3.3 of Bohm and Wilking). Assume that there exists a pinching family () starting at the cone of nonnegative curvature operators. Then, the statement of theorem 1 is true, i.e., there exists a pinching family starting at the cone of 2-nonnegative curvature operators.
Proof. Note that the theorem 1 is well-known in dimension 3 (see the example 1). Therefore, we can suppose that .
Since the property of a cone being closed convex -invariant and invariant by the ODE (1) is closed by intersections (exercise), given a pinching family , starting at the cone of nonnegative curvature operators, it suffices to show that where , and is the cone of 2-nonnegative curvature operators. Indeed, in this situation, we can extend the family ( and ) to a pinching family starting at by defining it on the second portion of the ”time” interval as a reparametrization of the pinching family (here we are using the closedness by intersection of the ODE-invariance property quoted in the beginning of the proof).
On the other hand, to see that , we recall the definition of is
where . Fix a 2-nonnegative curvature operator. We know from standard estimates that the smallest eigenvalue of verifies and, since is 2-nonnegative, the smallest eigenvalue of is . Hence, in order to show that (i.e., is a nonnegative curvature operator), it suffices to see that
By the definition of , we obtain , so that the previous inequality is valid (for ). This completes the proof of the lemma.
At this stage, it remains to exhibit a pinching family starting at the cone of nonnegative curvature operators. To do so, we will use the following two subfamilies described in the next two lemmas:
Lemma 2 (lemma 3.4 of Bohm and Wilking). Given , define
Then, the cone is invariant by the vector-field .
Lemma 3 (lemma 3.5 of Bohm and Wilking). Fix and for a given put
Then, the cone is invariant by the vector-field .
Assuming (momentarily) these two lemmas, we can show the theorem 2:
Proof of theorem 2 (modulo lemmas 2 and 3). Firstly we apply lemma 1 to reduce the proof of theorem 2 to the existence of a pinching family of cones starting at the cone of nonnegative curvature operators. Secondly, we note that the concatenation of the families of cones provided by lemmas 2 and 3 is a pinching family starting at the cone of nonnegative curvature operators (after a reparametrization of , of course). In fact, it is easy to see that the family of cones of lemma 2 starts at the (). Also, these two families of cones coincide at the parameters and (so that it is possible to make the concatenation). Finally, for the curvature operators belonging to the cones provided by lemma 3, we have (so that these cones converges to ). Combining these three facts, we obtain our desired pinching family (starting at ), so that the proof of theorem 2 is complete.
Remark 4. The construction of the pinching family using both lemmas 2 and 3 is necessary because the family of cones provided by lemma 2 is not a pinching family. In fact, although this family is invariant by the ODE (1) for all , it converges to the cone of Einstein curvature operators (instead of ).
The proofs of the lemmas 2 and 3 have a very similar flavor. In particular, we will not repeat ourselves so that we will discuss only the proof of lemma 3 (for the details about the lemma 2 see the proof of lemma 3.4 of Bohm and Wilking).
Proof of lemma 3. It suffices to show that the pull-back of the vector-field lies on the interior of the tangent cone where , and
To do so, we begin by showing that is positive definite. Since we are assuming that is positive semi-definite, it is sufficient to prove that . In this direction, we look at the explicit formula of the eigenvalues of from corollary 1. Using the definition of , we see that the formula of simplifies to:
From the assumption , we have that . In particular, . Plugging this information into the previous formula, we get (for ):
Hence, is positive. Next, we complete the proof of lemma 3 by showing that preserves the Ricci pinching condition (appearing in the definition of ). Firstly we note that the definition of combined with the formula of corollary 1 for the eigenvalues of the traceless Ricci tensor of gives
Putting together this expression with the previous formula for and the eigenvalue estimate (coming from the fact ), it is not hard to see that our task is to prove that
Since , we can neglet the terms containing . Dividing by the resulting expression, we obtain
Since this last inequality is obvious, the proof of the lemma 3 is complete.
Now we will briefly indicate in the next section how to apply the theorem 2 in order to conclude Bohm and Wilking theorem.
–Application of the pinching families to the proof of Bohm and Wilking theorem–
Recall that the basic idea of the proof of Bohm and Wilking theorem is that the existence of a pinching family starting at the cone of 2-nonnegative curvature operators forces the ODE (1) associated to the Ricci flow to evolve the initial metric to a limit metric of constant sectional curvature (i.e., a ”round” metric). However, this is not quite immediate from the definition of pinching family essentially because the convergence of the initial metric to a round metric depends also on good bounds for some geometric quantity (such as the scalar curvature) in order to avoid a possible scenario where the (curvature operator of the) metric approaches the cone and (at the same time) it blows up. Fortunately, the exclusion of this bad scenario can be done by standard ODE arguments which can be roughly described as follows. Let be a pinching family starting at the cone of 2-nonnegative curvature operators. By compactness of the manifold , the (2-positive) curvature operator of the initial metric satisfies for a large constant and a small constant . It turns out that elementary arguments from the theory of ODE shows that this last property of implies the existence of an invariant set containing such that is essentially contained in for all (the word essentially here means that is relatively compact). For the details of the construction of , see the section 4 of the paper of Bohm and Wilking. Once we have the set with the previous properties, we can use Shi’s derivative estimates to get a priori bounds for the curvature operators of the metrics evolving by the Ricci flow. Moreover, one can show finite time extinction (at time say) of the Ricci flow. Since the Ricci flow exists as long as the curvature stays bounded, if we rescale by a factor so that the maximum of the sectional curvatures of is 1, we get as . This implies that the limit metric of the Ricci flow has curvature operator inside the set
In other words, the limit metric has constant sectional curvature, so that the proof of the theorem of Bohm and Wilking is complete.
Of course, there are some minor details left behind this argument but it is morally correct. Nevertheless, the reader can see that the previous argument has nothing to do with a pinching family starting at the cone of 2-positive operators but it works ipsis-literis for the case of an arbitrary pinching family. In other words, the previous paragraph provides a sketch of proof of the following result:
Theorem 3 (theorem 5.1 of Bohm and Wilking). Let be a pinching family and be a Riemannian manifold whose curvature operator lies inside the interior of . Then, the (normalized) Ricci flow evolves to a limit metric of constant sectional curvature.
For a more complete discussion of this theorem, see the original article of Bohm and Wilking.
At this point, we end our discussion of Bohm and Wilking method of deformation of curvature conditions which are invariant by the ODE (1) (and a fortiori by the Ricci flow).