We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. In this lecture, we will show (at a heuristic level only) how the monotonicity of Perelman’s reduced volume can also be “derived”, in a formal sense, from another source of monotonicity, namely the relative Bishop-Gromov inequality in comparison geometry (which has already been alluded to in previous lectures). Interestingly, in order to obtain this connection, one must first reinterpret parabolic flows such as Ricci flow as the limit of a certain high-dimensional Riemannian manifold as the dimension becomes infinite; this is part of a more general philosophy that parabolic theory is in some sense an infinite-dimensional limit of elliptic theory. Our treatment here is a (liberally reinterpreted) version of Section 6 of Perelman’s paper.
In the next few lectures we shall give a rigorous proof of this monotonicity, without using the infinite-dimensional limit and instead using results related to the Li-Yau-Hamilton Harnack inequality. (There are several other approaches to understanding Perelman’s reduced volume, such as Lott’s formulation based on optimal transport, but we will restrict attention in this course to the methods that are in Perelman’s original paper.)
— The Bishop-Gromov inequality —
Let p be a point in a complete d-dimensional Riemannian manifold . As noted in Lecture 7, we can use the exponential map to pull back M and g to the tangent space , which is also equipped with the radial variable r and the radial vector field . From Exercise 7 of Lecture 7, we have the transport equation
for the volume measure , and a transport inequality
for the Laplcian which appears in (1). In particular, if we assume the lower bound
for Ricci curvature in a ball for some real number , then from the Gauss lemma (Lemma 1 of Lecture 7) we have
Also, from an expansion around the origin (see e.g. (13) or (15) from Lecture 7) we have
for small r. In principle, (4) and (5) lead to upper bounds on , which when combined with (1) lead to upper bounds on , which in turn lead to upper bounds on . One can of course just go ahead and compute these bounds, but one computation-free way to proceed is to introduce the model geometry , defined as
- the standard round sphere of radius (and thus constant sectional curvature K) if (Example 1 from Lecture 7);
- the standard hyperbolic space of constant sectional curvature K if (Example 2 from Lecture 7); or
- the standard Euclidean space if K=0.
As all of these spaces are homogeneous (in fact, they are symmetric spaces), the choice of origin p in this model geometry is irrelevant. Observe that the orthogonal group acts isometrically on each of these spaces, with the orbits being the spheres centred at p. This implies that at any point q not equal to p, is invariant under conjugation by the stabiliser of that group on q, which easily implies that it is diagonal on the tangent space to the sphere (i.e. to the orthogonal complement of ). From this we see that for this model geometry, the inequality in (2) is in fact an equality. Since the model geometry also has constant sectional curvature K (which implies equality in (3)), we thus see that one has equality in (4) for this model geometry as well. From this we can conclude:
Lemma 1. (Relative Bishop-Gromov inequality) With the assumptions as above, the volume ratio is a non-increasing function of r as .
Exercise 1. Prove Lemma 1. (Hints: One can avoid all issues with non-injectivity by working inside the cut locus of p, which determines a star-shaped region in . In the positive curvature case K > 0, the model geometry has a finite radius of injectivity, but observe that we may without loss of generality reduce to the case when is less than or equal to that radius (or one can invoke Myers’ theorem, see Exercise 2 below). To prove the monotonicity of ratios of volumes of balls, it may be convenient to first achieve the analogous claim for ratios of volumes of spheres, and then use the Gauss lemma and the fundamental theorem of calculus to pass from spheres to balls.)
Exercise 2. Prove Myers’ theorem: if a Riemannian manifold obeys (3) everywhere for some , then the diameter of the manifold is at most . (Hint: in the model geometry, the sphere of radius r collapses to a point when r approaches .)
Remark 1. This Lemma implies the volume comparison result whenever one has bounded normalised curvature, which was used in the previous lecture; indeed, thanks to the above inequality, it suffices to prove the claim for model geometries.
Setting K=0, we obtain
Corollary 1. Let be a complete d-dimensional Riemannian manifold of non-negative Ricci curvature, and let p be a point in M. Then is a non-increasing function of r.
Let us refer to the quantity as the Bishop-Gromov reduced volume at the point p and the scale r; thus we see that this quantity is dimensionless (i.e. invariant under scaling of the manifold and of r), and non-increasing in r when one has non-negative Ricci curvature (and in particular, for Ricci-flat manifolds).
Exercise 3. Use the Bishop-Gromov inequality to state and prove a rigorous version of the following informal claim: if a Riemannian manifold is non-collapsed at a point p at one scale (as defined in Lecture 7), then it is also non-collapsed at all larger scales .
— Parabolic theory as infinite-dimensional elliptic theory —
We now come to an interesting (but still mostly heuristic) correspondence principle between elliptic theory and parabolic theory, with the latter being viewed as an infinite-dimensional limit of the former, in a manner somewhat analogous to that of the central limit theorem in probability. To get some idea of what I mean by this correspondence, consider the following (extremely incomplete, non-rigorous, inaccurate, and imprecise) dictionary:
|Riemannian manifold (M,g)||Riemannian flow|
|Complete manifold||Ancient flow of complete manifolds|
|Spatial origin 0||Spacetime origin|
|Elliptic scaling||Parabolic scaling|
|Laplace equation||Heat equation|
|Ricci flat manifold||Ricci flow|
|Mean value principle||Fundamental solution|
|Normalised measure on the sphere||Heat kernel|
|Maximum principle||Maximum principle|
|Ball of radius O(r) around spatial origin||Cylinder of radius O(r) and height extending backwards in time from spacetime origin|
|Radial variable r=|x||||x| or|
|Bishop-Gromov reduced volume||Perelman reduced volume|
Remark 2. Of course, we have not defined Perelman reduced volume yet, but the point is that the monotonicity of Perelman reduced volume for Ricci flow is supposed to be the parabolic analogue of the monotonicity of Bishop-Gromov reduced volume for Ricci-flat manifolds. Note that one has two competing notions of the parabolic radial variable, |x| and , where is the backwards time variable; the ratio between these two competitors is essentially the Perelman reduced length, which does not really have a good analogue in the elliptic theory (except perhaps in the “latitude” variable one gets when decomposing a sphere into cylindrical coordinates).
It is well known that elliptic theory can be viewed as the static (i.e. steady state) special case of parabolic theory, but here we want to discuss a rather different connection between the two theories that goes in the opposite direction, in which we view parabolic theory as a limiting case of elliptic theory as the dimension d goes to infinity.
To motivate how this works, let us begin with a smooth ancient solution to the Euclidean heat equation
and ask how to convert it to a high-dimensional solution to the Laplace equation. At first glance this looks unreasonable: the Laplacian only contains second order derivative terms, but we have to somehow generate the first-order derivative out of this. The trick is to use polar coordinates. Recall that if we parameterise a Euclidean variable away from the origin as for and , then the Laplacian of a function can be expressed by the classical formula
where is the Laplace-Beltrami operator on the sphere. In particular, if f is a radial or spherically symmetric function (so by abuse of notation we write ), we have
Now if we look at the high-dimensional limit (noting that f, being radial, is well defined in every dimension), we see that the first order term dominates, despite the fact that is a second order operator. To clarify this domination (and to bring into view the operator appearing in (6)), let us make the change of variables
(thus is the average of the squared coordinates ). A quick application of the chain rule then yields
(one can also see this by writing and applying the Laplacian operator directly). If we restrict attention to the region of where all the coordinates are O(1), so and , and fix while letting N go off to infinity, we thus see that converges to (with errors that are ).
Returning back to our ancient solution to the heat equation (6), it is now clear how to express this solution as a high-dimensional nearly harmonic function: if we define the high-dimensional lift of u to the N+d-dimensional Euclidean space for some large N by using the change of variables (9), i.e.
then we see from (10) and (6) that is nearly harmonic as claimed; indeed we have
in the region , which implies as before that
Remark 3. Writing y in polar coordinates as , the metric on can be expressed as
where is the metric on the sphere of constant curvature . This expression is essentially the first equation in Section 6 of Perelman’s paper in the Euclidean case. Perelman works exclusively in polar coordinates, but I have found that the Cartesian coordinate approach can be more illuminating at times.
Remark 4. The formula (9) seems closely related to Itō’s formula from stochastic calculus, combined perhaps with the central limit theorem, though I was not able to make this connection absolutely precise. Note that for reasons of duality, stochastic calculus tends to involve the backwards heat equation rather than the forwards heat equation (see e.g. the Black-Scholes formula), which seems to explain why the minus sign in (9) is not present in Itō’s formula.
To illustrate how this correspondence could be used, let us heuristically derive the classical formula
for solutions to the heat equation (6) from the classical mean value principle
for harmonic functions . Actually, it will be slightly simpler to use the mean value principle for balls rather than spheres,
though in high dimensions there is actually very little difference between balls and spheres (the bulk of the volume of a high-dimensional ball is concentrated near its boundary, which is a sphere).
Let and be as in (6) and (11). From (12) we see that is almost harmonic; let us be non-rigorous and pretend that is close enough to harmonic that the formula (17) remains accurate for this function. We write the volume of the ball as for some constant . As for the integrand in (17), we use polar coordinates , and rewrite (17) as
for some other constant . In view of (9), it is natural to write for some , and in view of (13) it is natural to work in the regime in which , and . Because is so rapidly increasing when N is large, the bulk of the inner integral is concentrated at its endpoint (cf. our previous remark about high-dimensional balls concentrating near their boundary), and so we expect
Since is so much larger than |x| in our regime of interest, we can heuristically approximate by . Also, by Taylor approximation we have
Putting all this together, and substituting , we heuristically conclude
for some other constant . Taking limits as we heuristically obtain (15) up to a constant.
Exercise 4. Work through the calculations more carefully (but still heuristically), using Stirling’s approximation to the Gamma function, together with the classical formulae for the volume of balls and spheres, to verify that one does indeed get the right constant of in (15) at the end of the day (as one must).
Now let us perform a variant of the above computations which is more closely related to the monotonicity of Perelman’s reduced volume. The Euclidean space is of course Ricci-flat, and so from Corollary 1 we know that the Bishop-Gromov reduced volume
is non-decreasing in (and thus non-decreasing in ). (Of course, being Euclidean, (22) is equal to a constant ; but let us ignore this fact (which we have already used in our heuristic derivation of (15)) for now.) Repeating all the above computations (but with u and replaced by 1) we thus heuristically conclude that the quantity
is also non-decreasing in . (Indeed, this quantity is equal to for all .) The quantity (23) is precisely the Perelman reduced volume of Euclidean space (which we view as a trivial example of an ancient Ricci flow) at the spacetime origin (0,0) and backwards time parameter .
— From Ricci flow to Ricci flat manifolds —
We have seen how ancient solutions to the heat equation on a Euclidean spacetime can be viewed as (approximately) harmonic functions on an “infinitely high dimensional” Euclidean space. Now we would like to analogously view ancient solutions to a heat equation on a flow of Riemannian manifolds as harmonic functions on an “infinitely high dimensional” Riemannian manifold, and similarly to view ancient Ricci flows as infinite dimensional infinitely high dimensional Ricci-flat manifolds.
Let’s begin with the former task. Starting with an ancient flow of d-dimensional Riemannian metrics for (which we will not assume to be a Ricci flow just yet) and a large integer N, we can consider the N+d-dimesional manifold . As a first attempt to mimic the situation in the Euclidean case, it is natural to endow with the Riemannian metric given by the formula
where t is given by the formula (9). In terms of local coordinates, if we use the indices a,b,c to denote the d indices for the x variable and i,j,k to denote the N indices for the y variable, we have
where is the Kronecker delta. From this we see that the volume measure on is given by
and the Dirichlet form
for this Riemannian manifold is given by
where is the gradient of u in the x variable using the metric g(t). We can then integrate by parts to compute the Laplacian . Recalling that varies in t by the formula
(equation (19) from Lecture 1) and using (9) and the chain rule, we see that
where is the Laplace-Beltrami operator in the x variable using the metric g(t). If we specialise to radial functions
and use (10) and the chain rule, we can rewrite (29) as
Thus we see that if u solves the heat equation , then its lift is approximately harmonic in the sense that in the region where and x is confined to a compact region of space.
Remark 5. The term in (32) is somewhat annoying; we will later tweak the metric (24) in order to remove it (at the cost of other, more acceptable, terms).
Now let us see whether Ricci flows lift to approximately Ricci-flat manifolds . We begin by computing the Christoffel symbols in local coordinates, where refer to the N+d combined indices coming from the indices a on M and the indices i on . We recall the standard formula
for the Christoffel symbols of a general Riemannian manifold in local coordinates. Specialising to the metric (25), some computation reveals that
Now the Ricci curvature can be computed from the Christoffel symbols by the standard formula
If we apply this formula we obtain (after some computation)
We thus see that if the original flow obeys the Ricci flow equation , then the lifted manifold is nearly Ricci flat in the sense that all components of the Ricci curvature tensor are O(1/N) (in the region ). In fact the above estimates show that the Ricci curvature tensor is also O(1/N) in the operator norm sense and in the Hilbert-Schmidt (or Frobenius) sense.
It turns out that this approximation is not quite good enough for applications to Ricci flow, mainly because the term in (36) gives a significant contribution to the trace of the Ricci tensor (i.e. the scalar curvature ), even in the limit . It turns out however that one can eliminate this problem by adding a correction term to the metric (24) involving the scalar curvature. More precisely, given an ancient Ricci flow , define the modified metric by the formula
where of course is the derivative of the radial variable r, and R(t,x) is the scalar curvature of g(t) at x. In coordinates, we have
Exercise 5. Let be a smooth ancient Ricci flow on , and let be defined by (37). Show that in the region where (so ) and x ranges in a compact set, the Christoffel symbols take the form
and the Ricci curvature takes the form
Exercise 6. Let the assumptions and notation be as in Exercise 5, let be a smooth function, and let be as in (31). Show that the Laplacian associated to obeys a similar formula to (32), but with the term replaced by terms which are when t, x are bounded.
— Perelman’s reduced length and reduced volume —
In the previous discussion, we have converted a Ricci flow to a Riemannian manifold of much higher dimension which is almost Ricci flat. Let us adopt the heuristic that this latter manifold is sufficiently close to being Ricci flat that the Bishop-Gromov inequality (Corollary 1) holds (at least in the asymptotic limit ), thus the Bishop-Gromov reduced volume should heuristically be non-increasing in , where we fix a spatial origin .
In order to exploit the above heuristic, we first need to understand the distance function on . Let be a point in , and consider a length-minimising geodesic from to , where we have normalised the length of the parameter interval by the formula .
Observe that the metric (37) can be rewritten in polar coordinates (after substituting ) as
(which is essentially the first formula in Section 6 of Perelman’s paper). Note that the angular variable only influences the second term in this metric and not the other two. Because of this, one sees that the geodesic must keep constant in order to be length-minimising (i.e. for the duration of the geodesic). Turning next to the variable, we then see that for N large enough, the geodesic should increase continuously from 0 to (as the term in (41) will severely penalise any backtracking. After a reparameterisation we may in fact assume that increases at constant speed, thus we have
for some path from to . Using (41), the length of this geodesic is
which by Taylor expansion is equal to
where the -length of is defined as
Note that this quantity is independent of N. Thus, heuristically, geodesics in from to should (approximately) minimise the -length. If we define to be the infimum of over all paths from to , we thus obtain the heuristic approximation
Exercise 7. When M is the Euclidean space (with the trivial Ricci flow, of course), show that , and the minimiser is given by .
From (46) we see that the ball in of radius centred at (where, as always, we are in the regime , so ) should heuristically take the form
If we make the plausible assumption that varies smoothly in , then (47) is heuristically close (when N is large) to
Now, the volume measure of (37) is of the form , and so the volume of (49) is approximately
(Note there is a slight abuse of notation since depends on , but it will soon be clear that this abuse is harmless.) When N is large, the inner integral is dominated by its right endpoint as before, and so (50) is approximately
We can Taylor expand this to be approximately
where the Perelman reduced length is defined as
Example 1. Continuing the Euclidean example of Exercise 7, we have , which is the familiar exponent in the fundamental solution (15). This is, of course, not a coincidence.
From (52) we thus heuristically conclude that the Bishop-Gromov reduced volume of at and at radius is approximately equal to a constant multiple of , where the Perelman reduced volume is defined as
Example 2. Again continuing the Euclidean example, the reduced volume in Euclidean space (with the trivial Ricci flow) is always .
Formally applying Corollary 1, we are thus led to
Conjecture 1. (Monotonicity of Perelman reduced volume) Let be a Ricci flow on , and let . Then the quantity for is monotone non-increasing in .
Remark 6. Note here we are not taking the Ricci flow to be ancient; this would correspond to the manifold being replaced by an incomplete manifold, of radius about . However, because of the restriction , the above heuristic arguments never “encounter” the lack of completeness, and so it is reasonable to expect that the conjecture will continue to hold in the non-ancient case. This is of course an essential point for our applications, since the Ricci flows we study are not assumed to be ancient.
Remark 7. At an crude heuristic level, the Perelman reduced volume is roughly like (since, in view of Exercise 7, we expect to behave like , especially in regions of bounded normalised curvature, where we are deliberately vague about exactly what metric we using to define d). This heuristic suggests that Conjecture 1 should be able to establish the non-collapsing result we want (Theorem 2 from Lecture 7). This will be made more rigorous in subsequent lectures. For now, we observe that the Perelman reduced length and reduced volume are dimensionless (just as the Bishop-Gromov reduced volume is), which as discussed in Lecture 7 is basically a necessary condition in order for this quantity to force non-collapsing of the geometry.
As far as I am aware, there is no rigorous proof of Conjecture 1 that follows the above high-dimensional comparison geometry heuristic argument. Nevertheless, it is possible to prove Conjecture 1 by other means, and in particular by developing parabolic analogues of all the comparison geometry machinery that is used to prove the Bishop-Gromov inequality (and in particular, developing a theory of -geodesics analogous to the “elliptic” theory of geodesics on a Riemannian manifold. This will be the focus of the next few lectures.
Remark 8. It seems of interest to try to make the above arguments more rigorous, and to expand the dictionary between elliptic and parabolic equations. I do not know however of much literature in this direction, apart from Section 6 of Perelman’s original paper (see also Section 3.1 of Cao-Zhu), in which a few other parabolic notions (e.g. the backwards heat equation, or the modified Ricci flow from the previous lecture) are reinterpreted as high-dimensional elliptic notions. See however the work of Chow and Chu (see also this sequel paper), which views parabolic theory as a degenerate version of elliptic theory; Perelman’s viewpoint can be interpreted as a regularisation of Chow-Chu’s viewpoint.