The study of Riemannian manifolds with positive (scalar, Ricci and/or sectional) curvature is a vast and relatively old subject Differential Geometry. Among one of the classical (old) results in this area, one has the following theorem proved by H. Weyl in 1916:
Theorem 1 Let be a metric on with positive Gaussian curvature. Then, there exists a continuous path of metrics on with positive Gaussian curvature such that and is a round metric, i.e., has constant Gaussian curvature.
In fact, the proof of Weyl’s theorem is not especially long assuming the so-called uniformization theorem.
Proof: By the uniformization theorem, one knows that there exists a round metric in the conformal class of , i.e., is a round metric of the form .
On the other hand, a direct computation reveals that
where denotes the Gaussian curvature of the metric and is the Laplace-Beltrami operator associated to .
In particular, it follows that , , is a continuous path of metrics of positive Gaussian curvature on such that and is a round metric.
Remark 1 Note that H. Weyl’s theorem implies that the space of metrics of positive Gaussian curvature on is path-connected. Actually, as it was shown by J. Rosenberg and S. Stolz, the space is contractible.
In this post, we’ll be interested in the natural question of generalizing Weyl’s theorem to higher dimensions. Here, it is worth to point out that a naive (straightforward) extension of Weyl’s argument is simply not going to work because, in general, there is no analog of the uniformization theorem in higher dimensions (see, e.g., this discussion in MathOverflow of the 4-dimensional case).
During this post, we will use the following notations. Given a compact orientable manifold , we will denote by the set of Riemannian metrics on with positive scalar curvature (psc for short), and by the moduli space of psc (positive scalar curvature) metrics on . In simple terms, the moduli space is the set of psc (positive scalar curvature) metrics modulo isometries.
Just to give a flavor on how delicate it might be to generalize Weyl’s theorem to higher dimensions, let us mention that, after the works of N. Hitchin, R. Carr, M. Gromov and B. Lawson (see here and here), M. Kreck and S. Stolz, it is known that the moduli space of psc metrics on spin manifolds might be disconnected. For instance:
Theorem 2 (M. Kreck and S. Stolz) For , the moduli space
has infinitely many connected components.
This result says that, starting by the -dimensional sphere , one has several higher-dimensional spheres supporting lots of “exotic metrics of positive scalar curvature” (in the sense that these metrics are not in the connected component of the standard round metric). Very roughly speaking, the basic idea behind this theorem is the fact that the obstructions for two metrics of positive scalar curvature to belong to the same connected component come from spin geometry or, more precisely, from the index theory of the Dirac operator.
The goal of the post is the discussion of a result of Fernando Codá Marques saying that Weyl’s theorem admits a nice generalization in not-so-high dimensions. More precisely, in this paper here, Fernando shows that:
In this post, we will closely follow the original article of Fernando to outline (in the next sections) a proof of this theorem based on G. Perelman’s Ricci flow with surgery and the conformal method.
Remark 2 If I remember correctly, Fernando first heard about the statement of his theorem as a question during some lectures given by Richard Schoen in 2006.
Let us mention that, by putting together his Theorem 3 with a theorem of J. Cerf ensuring that is path-connected (and, furthermore, its homotopy type is known after this work of A. Hatcher, Fernando gets the following corollary:
Corollary 4 (F. C. Marques) The space of positive scalar curvature metrics on the -sphere is path-connected.
Remark 3 In Section 9 of his article, Fernando gives an application of this corollary to General Relativity, but we will not discuss today this aspect of Fernando’s paper.
Closing this introduction, let us make the following useful definition/notation (for later use):
with and .
1. Ricci flow with surgery
The basic idea behind Theorem 3 is very simple: we’ll try to connect a psc metric to some canonical (“model”) metrics.
In dimensions, the prototypical situation where this idea was successfully used is R. Hamilton’s 1982 theorem saying that, given a compact -manifold with a metric with positive Ricci curvature, if one looks at the solution of the so-called Ricci flow
then the metrics have positive Ricci curvature and, furthermore, certain rescalings of (corresponding to the solution of the so-called normalized Ricci flow) converge to a round (constant sectional curvature metric) metric . In particular, it follows that the moduli space of positive Ricci curvature metrics on is path-connected (and, actually, contractible). Moreover, since is the unique simply connected -manifold admitting a metric with positive sectional curvature, we have that where is a discrete group of isometries of a round metric on .
In other terms, in the case of quotients of the sphere , the round metrics are canonical metrics, and all other metrics with positive Ricci curvature can be connected to these canonical metrics.
In general, we want to repeat this strategy for general psc metrics in general compact -manifolds. Of course, the first difficulty we encounter is the fact that, for an arbitrary -manifold, the notion of canonical/model metric is vague, even though, for example, one is certainly willing to say that metrics enjoying some kinds of symmetries (such as the round metrics on ) are canonical metrics. Since the notion of symmetry is sensitive to the underlying manifold , it is a good idea to start our considerations by the following question:
What are the orientable -manifolds admitting a psc metric (or, equivalently, such that )?
The answer to this question was given by G. Perelman using his Ricci flow with surgery and his description of the singularities developed by this flow. Since the Ricci flow with surgery is a crucial tool for the proof of Theorem 3, we will briefly revisit some of its features in the next subsection.
Disclaimer. Our discussion of the Ricci flow with surgery will be very informal. In particular, if the reader wishes to consult more responsible sources/expositions about this flow (with plenty of details), we recommend consulting the book of J. Morgan and G. Tian, this paper of B. Kleiner J. Lott, and/or these lecture notes of Terence Tao.
1.1. Ricci flow with surgery I
The existence of short-time solutions of the Ricci flow was proved by R. Hamilton in his 1982 paper quoted above. However, as Hamilton’s theorem on metrics with positive Ricci curvature hints, it is usually interesting to know whether the Ricci flow has long-time solutions in order to be able to connect a given metric to a canonical metric . In the context of positive Ricci curvature, the existence of long-time solutions was proved by R. Hamilton’s 1982 paper, but, in the setting of psc metrics, it was known among the experts that the Ricci flow could become extinct or develop singularities in finite time.
More concretely, it is known that the the scalar curvatures of a solution of the Ricci flow on a -dimensional manifold , , satisfy the following evolution equation:
In particular, since , one can applies the maximum principle to deduce that, if (i.e., is psc), then (i.e., is psc) for all . Actually, this simple argument showing that the psc property is preserved by the Ricci flow can be improved to show that the Ricci flow associated to a psc metric ends in finite time. Indeed, since , one can use the maximum principle to get that, if , then
Since the quantity blows up (approaches ) as , we have that the solution of the Ricci flow can’t exist for , i.e., the Ricci flow solution becomes extinct in finite time.
Furthermore, it may happen that the Ricci flow starting at a psc metric develops a singularity in finite time. Here, the idea is that the initial metric may look like the following one in :
In this picture, we have two round spheres and glued along a very small neck . The presence of the small neck makes that the region around has a very large scalar curvature (compared with the round spheres and ), and thus the Ricci flow in dimensions has the “tendency” to degenerate (“pinch off”) the region near faster than the round spheres and . The reader will find more details on the formation of this kind singularity of the Ricci flow in this article here, for example.
Remark 4 The scenario described above doesn’t occur in dimension . Indeed, starting with two round -spheres and connected by a neck, the Ricci flow in dimension will actually uniformize this geometrical object in the sense that, up to rescaling, the Ricci flow will “undo” the neck and the geometry will approach the one of a round metric on (see, e.g., the figure in this link here). Actually, as it was shown by R. Hamilton, and X. Chen, P. Liu and G. Tian, the Ricci flow in dimension can be used to give a modern proof of the uniformization theorem.
At this stage, G. Perelman’s idea is the following. Using a very precise description of singularities developed by Ricci flow on -manifolds, he performs a surgery (that is “canonical” in terms of certain parameters) on in order to remove the regions with large curvature, then he runs again the Ricci flow in the remaining components of after surgery, and he continues this process of successive applications of surgeries and Ricci flows ad infinitum (or at least until the Ricci flow doesn’t become extinct). In the next subsection we will describe the Ricci flow with surgery with a little bit more of details, but, for now, let us explain a classification result of -manifolds admitting psc metrics following from G. Perelman’s work on the Ricci flow with surgery.
We start by recalling that the “simplest” examples of compact -manifolds admitting psc metrics are:
- and its quotients , and
Also, let us recall that, after the work of M. Gromov and B. Lawson, it is known that the psc property is stable under connected sums. Roughly speaking, given two -manifolds and with two psc metrics and , it is possible equipped the connected sum with a psc metric (GL standing for Gromov-Lawson) as follows. Firstly, one removes small balls and of and of radius centered at points , and, secondly, one attach to an adequate cylindrical region (neck) carrying a rotationally symmetric metric in a large portion of it. See the picture below.
Of course, the actual process – called Gromov-Lawson procedure – depends on parameters (e.g., for the radius of the balls and , and for the width of the cylindrical neck), and thus the Gromov-Lawson procedure provides a family of psc metrics on .
However, for our purposes, we don’t have to bother about specific choices of parameters because it is possible to check that all metrics obtained by this procedure are isotopic (in the sense of Definition 5).
Also, for later use, we will need the following nice feature of the Gromov-Lawson procedure under isotopies (again in the sense of Definition 5). Given and , one has that . In other terms, this means that it is possible to “follow” the Gromov-Lawson procedure under isotopies, and this is so because of the explicit nature of the Gromov-Lawson procedure as described in the original article.
In any case, once we know the basic types of -manifolds carrying psc metrics (namely and ) and we have that psc is stable under connected sums, one can start wondering about other -manifolds with psc metrics. Using his Ricci flow with surgery, G. Perelman gave the following answer to this question:
Theorem 6 (G. Perelman) Let be an orientable compact -manifold. If admits psc metrics (i.e., ), then is diffeomorphic to a connected sum of some quotients of spheres , , and some copies of .
Of course, this result put us in good shape to formally define canonical metrics on orientable -manifolds with : by writing
we declare that the canonical metrics of are those obtained by taking round metrics on and , products of round metrics on , and applying the Gromov-Lawson procedure. Pictorially, a canonical metric is
Remark 5 The factor in the connected sum above could be removed as it doesn’t change the topology of . However, we write this extra factor because it will serve as a sort of “reference piece” of as far as the proof of Theorem 3 is concerned.
By a result of H. de Rham, the canonical metrics on and/or are all isotopic to each other. This suggests that the proof of Theorem 3 could pass by the idea of trying to connect a given psc metric on a -manifold (without losing the psc property, of course!) to some connected sum of canonical metrics.
As we mentioned above, the key tool to connect arbitrary psc metrics to connected sums of canonical metrics is the Ricci flow with surgery of G. Perelman. For this reason, we will briefly review in the next subsection the main features of this flow needed for the proof of Theorem 3.
1.2. Ricci flow with surgery II
The Ricci flow with surgery of G. Perelman is a discontinuous process with the following properties. Starting with an orientable compact -manifold with a metric , we have a family , , such that
- for each , the family , , is a solution of the standard Ricci flow that becomes singular as ;
- is obtained from the preceding Ricci flow by a surgery (involving a certain number of parameters);
- the set of singular times is discrete.
Roughly speaking, the parameters in G. Perelman scheme are (almost rotationally symmetric) necks in the manifold where the curvature gets very large (as indicated in the first figure of this post).
Once G. Perelman sees such necks in the manifolds for very close to , his surgery procedure is the following: he cuts the necks in the middle and he attach (standard rotationally symmetry) caps as it is indicated in the figure below. In this way, he obtains a possibly disconnected Riemannian manifold that he uses as a new initial condition for the (standard) Ricci flow.
An interesting feature of G. Perelman’s scheme is that he only uses his surgery process to get rid/discard of regions of large curvature.
For our purposes, another nice property of the Ricci flow with surgery (proved independently by G. Perelman, and T. Colding and W. Minicozzi) is the following. Given is a psc metric, then is a psc metric for all and , and the flow becomes extinct in finite time for some in the sense that , i.e., by performing G. Perelman’s surgery procedure at time , one ends up by discarding everything.
This finite-extinction result for the Ricci flow with surgery is important in G. Perelman’s solution of Poincaré’s conjecture. Indeed, the idea is that, if you understand how the flow ends, then you can reason backwards to understand the topology of .
However, this description of Ricci flow with surgery for psc metrics is not sufficient for the proof of Theorem 3 because our final goal is to control the geometry of . In other terms, we need to understand the geometry of the so-called canonical neighborhoods, that is, the necks and caps involved in Perelman’s surgery process.
As we already indicated, the canonical neighborhoods are associated to regions of large curvature (in the sense that, if the scalar curvature of at a point is very large, say where is a small parameter, then belongs to a canonical neighborhood), and there are 3 types of them:
- (1) a whole component with positive sectional curvature;
- (2) a -neck , i.e., a small region with large curvature that after rescaling becomes “-close” to ;
- (3) a -cap, i.e., a small region with large curvature “-close” to the region obtained by adding a rotationally symmetry cap (with “-controlled geometry”) to an -neck.
We observe that the canonical neighborhoods of the type in item (1) above will disappear by the standard Ricci flow by the results in R. Hamilton’s 1982 paper. In particular, we will forget about this type of canonical neighborhoods and we will focus exclusively on -necks and -caps.
After this sketchy review of the geometry properties of the Ricci flow with surgery, we are ready to start the proof of Theorem 3.
2. General idea of the proof of Theorem 3
Let be an orientable compact -manifold admitting psc metrics, say
Given a psc metric on , we wish to reduce the proof of Theorem 3 to the usage of the Ricci flow with surgery to connect to some canonical metric on . Of course, this strategy only works if one can show that the set of canonical metrics on is path-connected, a fact that is not obvious at all!
So, let us start by justifying the path-connectedness of the set of canonical metrics via the so-called conformal method.
2.1. Conformal method
We claim that the connected sums of canonical metrics on and via the Gromov-Lawson procedure are isotopic to a single canonical metric. Indeed, let us start with the orientable compact -manifold equipped with the canonical metric shown in the left-hand side of the picture below.
Now, let us temporarily forget about the necks (see the right-hand side of the picture above), and let us focus on the spherical components glued together by a Gromov-Lawson procedure as indicated in the right-hand side of the figure above. The Gromov-Lawson procedure connecting the spherical components provides a rotationally symmetric psc metric, and, a fortiori, a locally conformally flat psc metric (as any rotationally symmetric metric is locally conformally flat) on the connected sum of the spherical components.
Here, even though there is no general uniformization theorem in dimension , the so-called conformal method of N. Kuiper, R. Schoen and S.-T. Yau says that locally conformally flat psc metrics are conformal, and thus isotopic, to round metrics on and/or ‘s.
Finally, since the Gromov-Lawson procedure is local, we can put the “forgotten” necks back to deduce that the canonical metrics on are all isotopic to each other.
After this preparatory step, the proof of Theorem 3 was really reduced to connect psc metrics to some canonical metric on via the Ricci flow with surgery.
2.2. Connecting psc metrics to canonical metrics
Let be a psc Riemannian orientable -manifold and denote by , , , the corresponding Ricci flow with surgery. Recall that are psc metrics and the flow becomes extinct in finite time, i.e., for some .
In this subsection, we will essentially follow the backward induction argument of G. Perelman in his proof of Poincaré’s conjecture. More precisely, for each , let us consider the following assertion:
- () every connected component of is isotopic to a canonical metric.
The backward induction argument consists into showing the following two claims:
- if holds, then holds.
Let us show that holds. By definition of , we have that , and hence has uniformly large curvature at all points for sufficiently close to the extinction time . In other words, we get that is a union of canonical neighborhoods.
As we said, the canonical neighborhoods have 3 types, but we can focus exclusively on -necks and -caps because the components with positive sectional curvatures will approach round metrics and disappear (by R. Hamilton’s results).
Consider some -neck or, more generally, some chain of -necks. The first possibility is that, by extending a chain of -necks, it starts to “go around” and then closes at some point to form a component. The second possibility is that, by extending a chain of -necks, we meet at some point a -cap. Then, since there is nothing left for the other side of the chain of necks to close up, it has to meet a cap on the other side also, so that this component is a sphere . The figure below illustrates these two possibilities.
Logically, there is no other possibility, so that our task is reduced to show that we can find an isotopy between the metric on these and components.
We begin with the case. Here, we have two subcases.
The first one occurs when the two caps meeting the chain of necks are -caps, i.e., both caps are -close to a rotationally symmetric -controlled cap added to a -neck. In this situation, one can deform “by hand” the metric to a rotationally symmetric (cigar-like) metric on . Then, since rotationally symmetric metrics are locally conformally flat, the conformal method allows to uniformize it (by isotopy) to a round metric on , and thus we are done.
The second subcase occurs when one of the caps is a -cap, but the other cap is a region where the sectional curvature is positive. In this situation, one can perform a Perelman’s surgery and then glue -caps on both sides.
Remark 6 This surgery is not done in G. Perelman’s work on Poincaré conjecture: indeed, this surgery is irrelevant from the topological point of view, but, as we’ll see below, Fernando was “forced” to make this new step in order to control the geometry of .
In this way, starting from our component , after surgery we get a new component carrying a metric with positive sectional curvature (this was not clearly stated above, but Perelman’s surgery preserves the positivity of sectional curvatures) and a component in the setting of the first subcase (i.e., obtained by attaching to -caps to a chain of necks). By Hamilton’s results, the standard Ricci flow on the positive sectional curvature component provides an isotopy to a round metric on , while the subcase of the component was already treated. However, we’re not directly interested in the geometry of the components after surgery (i.e., the geometry of and ), but rather about the geometry of the component before surgery.
At this point, the crucial remark is the following. Starting with a neck, if we do a surgery and then we undo the surgery via a Gromov-Lawson procedure, we obtain a metric that is isotopic to the original one! In other words, up to isotopy, the Gromov-Lawson procedure is a sort of inverse of G. Perelman’s surgery!
From this remark, one can show that the original component is isotopic to a round sphere as follows. By the remark, is isotopic to the Gromov-Lawson procedure on the components and , and we know that and are isotopic to a round . It follows that is isotopic to a Gromov-Lawson procedure on two round spheres, and, by the conformal method, we have that is isotopic to a round .
In any event, this completes the analysis of the case.
Now, let us pass to the case of a component . In this setting, we choose any -neck and we perform a surgery on . By the remark above, we have that is isotopic to a Gromov-Lawson procedure on a component consisting of a chain of necks with two caps attached to it, as shown in the figure below (extracted from Figure 8.4 in Fernando’s paper).
Note that corresponds precisely to a case that we already treated. In particular, we know that is isotopic to a round . Since the Gromov-Lawson reverts the surgery up to isotopy, we deduce from this that the component is isotopic to a canonical metric on .
At this point, we treated all cases ( and ) and this proves that the assertion holds.
Closing the proof of Theorem 3, let us quickly describe (in three phrases) why the validity of implies the validity of . By inspecting the surgery process of G. Perelman, one can show that every connected component of is isotopic to a connected sum of components of and unions of canonical neighborhoods. By induction hypothesis (i.e., holds), we have that the connected components of are isotopic to canonical metrics, while our discussion above of the and cases showed that the canonical neighborhoods are isotopic to canonical metrics. Since the connected sum (Gromov-Lawson procedure) behaves well under isotopies, we conclude that the components of are isotopic to canonical metrics and this proves .