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 (M,g). As noted in Lecture 7, we can use the exponential map to pull back M and g to the tangent space T_p M, which is also equipped with the radial variable r and the radial vector field \partial_r = \hbox{grad}(r). From Exercise 7 of Lecture 7, we have the transport equation

{\mathcal L}_{\partial_r} d\mu = (\Delta r)\ d\mu (1)

for the volume measure d\mu, and a transport inequality

\displaystyle \nabla_{\partial_r} \Delta r + \frac{1}{d-1} (\Delta r)^2 \leq \nabla_{\partial_r} \Delta r + |\hbox{Hess}(r)|^2 = - \hbox{Ric}(\partial_r, \partial_r) (2)

for the Laplcian \Delta r which appears in (1). In particular, if we assume the lower bound

\hbox{Ric} \geq (d-1) K g (3)

for Ricci curvature in a ball B(p,r_0) for some real number K, then from the Gauss lemma (Lemma 1 of Lecture 7) we have

\displaystyle \nabla_{\partial_r} \Delta r + \frac{1}{d-1} (\Delta r)^2 \leq -(d-1) K. (4)

Also, from an expansion around the origin (see e.g. (13) or (15) from Lecture 7) we have

\displaystyle \Delta r = \frac{d-1}{r} + O(r) (5)

for small r. In principle, (4) and (5) lead to upper bounds on \Delta r, which when combined with (1) lead to upper bounds on d\mu, which in turn lead to upper bounds on B(p,r_0). One can of course just go ahead and compute these bounds, but one computation-free way to proceed is to introduce the model geometry (M_K, g_K), defined as

  1. the standard round sphere \sqrt{K} \cdot S^d of radius \sqrt{K} (and thus constant sectional curvature K) if K > 0 (Example 1 from Lecture 7);
  2. the standard hyperbolic space \sqrt{-K} \cdot H^d of constant sectional curvature K if K < 0 (Example 2 from Lecture 7); or
  3. the standard Euclidean space {\Bbb R}^d 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 O(d) 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, \hbox{Hess}(r) 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 \partial_r). 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 \hbox{Vol}_{M,g}(B_{M,g}(p,r)) / \hbox{Vol}_{M_K,g_K}(B_{M_K,g_K}(p,r)) is a non-increasing function of r as 0 < r < r_0.

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 T_p M. In the positive curvature case K > 0, the model geometry M_K has a finite radius of injectivity, but observe that we may without loss of generality reduce to the case when r_0 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.) \diamond

Exercise 2. Prove Myers’ theorem: if a Riemannian manifold obeys (3) everywhere for some K > 0, then the diameter of the manifold is at most \pi/\sqrt{K}. (Hint: in the model geometry, the sphere of radius r collapses to a point when r approaches \pi/\sqrt{K}.) \diamond

Remark 1. This Lemma implies the volume comparison result \hbox{vol}(B(p,r))/\hbox{vol}(B(p,r/2)) = O(1) 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. \diamond

Setting K=0, we obtain

Corollary 1. Let (M,g) be a complete d-dimensional Riemannian manifold of non-negative Ricci curvature, and let p be a point in M. Then \hbox{Vol}(B(p,r))/r^d is a non-increasing function of r.

Let us refer to the quantity \hbox{Vol}(B(p,r))/r^d 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 r_0 > 0 (as defined in Lecture 7), then it is also non-collapsed at all larger scales r_1 > r_0. \diamond

— 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:

Elliptic Parabolic
Riemannian manifold (M,g) Riemannian flow t \mapsto (M,g(t))
Complete manifold Ancient flow of complete manifolds
Spatial origin 0 Spacetime origin (0,0)
Elliptic scaling x \mapsto \lambda x Parabolic scaling (t,x) \mapsto (\lambda^2 t, \lambda x)
Laplace equation \Delta u = 0 Heat equation -\partial_t u + \Delta u = 0
Ricci flat manifold \hbox{Ric} = 0 Ricci flow \partial_t g = - 2 \hbox{Ric}
Mean value principle u(0) = \int_{S^{d-1}} u(r\omega)\ d\mu(\omega) Fundamental solution u(0,0) = \frac{1}{(4\pi\tau)^{d/2}} \int_{{\Bbb R}^d} e^{-|x|^2/4\tau} u(-\tau,x)\ dx
Normalised measure on the sphere r \cdot S^{d-1} Heat kernel \frac{1}{(4\pi\tau)^{d/2}} e^{-|x|^2/4\tau}\ dx
Maximum principle Maximum principle
Ball of radius O(r) around spatial origin Cylinder of radius O(r) and height O(r^2) extending backwards in time from spacetime origin
Radial variable r=|x| |x| or \sqrt{-t} = \sqrt{\tau}
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 \sqrt{\tau}, where \tau := -t 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). \diamond

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 u: (-\infty,0] \times {\Bbb R}^d \to {\Bbb R} to the Euclidean heat equation

-\partial_t u + \Delta_x u = 0 (6)

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 \partial_t out of this. The trick is to use polar coordinates. Recall that if we parameterise a Euclidean variable y \in {\Bbb R}^N away from the origin as y = r \omega for r > 0 and \omega \in S^{N-1}, then the Laplacian \Delta_y f of a function f: {\Bbb R}^N \to {\Bbb R} can be expressed by the classical formula

\displaystyle \Delta_y f = \partial_{rr} f + \frac{N-1}{r} \partial_r f + \frac{1}{r^2} \Delta_\omega f (7)

where \Delta_{\omega} 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 f(y) = f(r)), we have

\displaystyle \Delta_y f = \partial_{rr} f + \frac{N-1}{r} \partial_r f. (8)

Now if we look at the high-dimensional limit N \to \infty (noting that f, being radial, is well defined in every dimension), we see that the first order term \frac{N-1}{r} \partial_r f dominates, despite the fact that \Delta_y is a second order operator. To clarify this domination (and to bring into view the operator -\partial_t appearing in (6)), let us make the change of variables

\displaystyle t = -\tau = - \frac{r^2}{2N} = -\frac{y_1^2 + \ldots + y_N^2}{2N} (9)

(thus \tau = -t is the average of the squared coordinates y_1^2,\ldots,y_N^2). A quick application of the chain rule then yields

\displaystyle \Delta_y f = -\frac{2t}{N} \partial_{tt} f - \partial_t f (10)

(one can also see this by writing f(y) = \tilde f(t) = \tilde f( -(y_1^2+\ldots+y_N^2)/2N ) and applying the Laplacian operator \Delta_y directly). If we restrict attention to the region of {\Bbb R}^N where all the coordinates y_i are O(1), so r^2 = O(N) and \tau = -t = O(1), and fix \tilde f while letting N go off to infinity, we thus see that \Delta_y f converges to - \partial_t f (with errors that are O(1/N)).

Returning back to our ancient solution u: (-\infty,0] \times {\Bbb R}^d \to {\Bbb R} 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 u^{(N)}: {\Bbb R}^N \times {\Bbb R}^d \to {\Bbb R} of u to the N+d-dimensional Euclidean space {\Bbb R}^N \times {\Bbb R}^d := \{ (y,x): y \in {\Bbb R}^N, x \in {\Bbb R}^d \} for some large N by using the change of variables (9), i.e.

\displaystyle u^{(N)}(y,x) := u(t,x) = u( - \frac{y_1^2+\ldots+y_N^2}{2N}, x ) (11)

then we see from (10) and (6) that u^{(N)} is nearly harmonic as claimed; indeed we have

\Delta_{y,x} u^{(N)} = \frac{r^2}{N^2} \partial_{tt} u - \partial_t u + \Delta_x u = O(1/N) (12)

in the region y_i = O(1), x = O(1), which implies as before that

y_i = O(1), x = O(1), r^2 = O(N), \tau = -t = O(1). (13)

Remark 3. Writing y in polar coordinates as y = r\omega, the metric ds^2 on {\Bbb R}^N \times {\Bbb R}^d can be expressed as

\displaystyle ds^2 = dr^2 + r^2 d\omega^2 + dx^2 = \frac{N}{2\tau} d\tau^2 + \tau d\omega_{1/2N}^2 + dx^2 (14)

where d\omega_{1/2N}^2 is the metric on the sphere S^N of constant curvature 1/2N. 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. \diamond

Remark 4. The formula (9) seems closely related to Itō’s formula dt = (dB)^2 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. \diamond

To illustrate how this correspondence could be used, let us heuristically derive the classical formula

\displaystyle u(0,0) = \frac{1}{(4\pi\tau)^{d/2}} \int_{{\Bbb R}^d} e^{-|x|^2/4\tau} u(-\tau,x)\ dx (15)

for solutions u: (-\infty,0] \times {\Bbb R}^d \to {\Bbb R} to the heat equation (6) from the classical mean value principle

\displaystyle u^{(N)}(0,0) = \frac{1}{\hbox{mes}(r \cdot S^{N+d-1})} \int_{S^{N+d-1}} u^{(N)}(r\omega)\ d\omega (16)

for harmonic functions u^{(N)}: {\Bbb R}^N \times {\Bbb R}^d \to {\Bbb R}. Actually, it will be slightly simpler to use the mean value principle for balls rather than spheres,

\displaystyle u^{(N)}(0,0) = \frac{1}{\hbox{Vol}(B^{N+d}(0,r_0))} \int_{|y|^2+|x|^2 \leq r_0^2} u^{(N)}(y,x)\ dy dx, (17)

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 u and u^{(N)} be as in (6) and (11). From (12) we see that u^{(N)} is almost harmonic; let us be non-rigorous and pretend that u^{(N)} is close enough to harmonic that the formula (17) remains accurate for this function. We write the volume of the ball B^{N+d}(0,r_0) as C_{N,d} r_0^{N+d} for some constant C_{N,d}. As for the integrand in (17), we use polar coordinates y = r \omega, dy = r^{N-1} dr d\omega and rewrite (17) as

\displaystyle c_{N,d} r_0^{-N-d} \int_{{\Bbb R}^d} \int_{0 \leq r \leq \sqrt{r_0^2-|x|^2}} u( -r^2/2N, x ) r^{N-1} dr dx (18)

for some other constant c_{N,d} > 0. In view of (9), it is natural to write r_0^2 = 2N \tau for some \tau > 0, and in view of (13) it is natural to work in the regime in which x = O(1), \tau = O(1), and r_0^2 = O(N). Because r^{N-1} 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

\displaystyle \int_{0 \leq r \leq \sqrt{r_0^2-|x|^2}} u( -r^2/2N, x ) r^{N-1} dr

\displaystyle \approx \frac{1}{N} (\sqrt{r_0^2-|x|^2})^N u( - (r_0^2-|x|^2)/2N, x ). (19)

Since r_0 is so much larger than |x| in our regime of interest, we can heuristically approximate u( - (r_0^2-|x|^2)/2N, x) by u( -r_0^2/2N,x) = u(-\tau,x). Also, by Taylor approximation we have

\displaystyle (\sqrt{r_0^2-|x|^2})^N \approx r_0^{N} \exp( - \frac{N |x|^2}{2 r_0^2} ). (20)

Putting all this together, and substituting r_0^2 = 2N\tau, we heuristically conclude

\displaystyle u(0,0) \approx \frac{\tilde c_{N,d}}{\tau^{d/2}} \int_{{\Bbb R}^d} e^{-|x|^2/4\tau} u(-\tau,x)\ dx (21)

for some other constant \tilde c_{N,d} > 0. Taking limits as N \to \infty 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 \frac{1}{(4\pi)^{d/2}} in (15) at the end of the day (as one must). \diamond

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 {\Bbb R}^N \times {\Bbb R}^d is of course Ricci-flat, and so from Corollary 1 we know that the Bishop-Gromov reduced volume

\displaystyle r_0^{-N-d} \int_{|y|^2+|x|^2 \leq r_0^2}\ dy dx (22)

is non-decreasing in r_0 (and thus non-decreasing in \tau). (Of course, being Euclidean, (22) is equal to a constant C_{N,d}; 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 u^{(N)} replaced by 1) we thus heuristically conclude that the quantity

\displaystyle \frac{1}{\tau^{d/2}} \int_{{\Bbb R}^d} e^{-|x|^2/4\tau}\ dx (23)

is also non-decreasing in \tau. (Indeed, this quantity is equal to (4\pi)^{d/2} for all \tau.) The quantity (23) is precisely the Perelman reduced volume of Euclidean space {\Bbb R}^d (which we view as a trivial example of an ancient Ricci flow) at the spacetime origin (0,0) and backwards time parameter \tau.

— 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 t \mapsto (M,g(t)) 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 t \mapsto (M,g(t)) of d-dimensional Riemannian metrics for t \in (-\infty,0] (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 M^{(N)} := {\Bbb R}^N \times M = \{ (y,x): y \in {\Bbb R}^N, x \in M \}. As a first attempt to mimic the situation in the Euclidean case, it is natural to endow M^{(N)} with the Riemannian metric g^{(N)} given by the formula

(dg^{(N)})^2 = dy^2 + dg(t)^2 (24)

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

g^{(N)}_{ab} = g_{ab}(t); g^{(N)}_{ai} = g^{(N)}_{ia} = 0; g^{(N)}_{ij} = \delta_{ij} (25)

where \delta is the Kronecker delta. From this we see that the volume measure d\mu^{(N)} on M^{(N)} is given by

d\mu^{(N)} = d\mu(t) dy (26)

and the Dirichlet form

E^{(N)}(u,v) := \int_{M^{(N)}} g^{(N)}( \nabla^{(N)} u, \nabla^{(N)} v )\ d\mu^{(N)}

= - \int_{M^{(N)}} \Delta^{(N)} u v\ d\mu^{(N)} (27)

= - \int_{M^{(N)}} u \Delta^{(N)} v\ d\mu^{(N)}

for this Riemannian manifold is given by

\displaystyle E^{(N)}(u,v) = \int_{{\Bbb R}^N} \int_{M(t)} \nabla_y u \cdot \nabla_y v + g(t)( \nabla_{x,g(t)} u, \nabla_{x,g(t)} v)\ d\mu(t) dy, (28)

where \nabla_{x,g(t)} u is the gradient of u in the x variable using the metric g(t). We can then integrate by parts to compute the Laplacian \Delta^{(N)} u. Recalling that d\mu(t) varies in t by the formula

\displaystyle \frac{d}{dt} d\mu(t) = \frac{1}{2} \hbox{tr}(\dot g) d\mu(t) (29)

(equation (19) from Lecture 1) and using (9) and the chain rule, we see that

\displaystyle \Delta^{(N)} u^{(N)} = \Delta_y u^{(N)} + \Delta_{x, g(t)} u^{(N)} - \frac{r}{2N} \hbox{tr}(\dot g) \partial_r u^{(N)} (30)

where \Delta_{x,g(t)} is the Laplace-Beltrami operator in the x variable using the metric g(t). If we specialise to radial functions

u^{(N)}(y,x) = u(t,x) (31)

and use (10) and the chain rule, we can rewrite (29) as

\displaystyle -\frac{t}{2N} \partial_{tt} u - \partial_t u + \Delta_{x,g(t)} u + \frac{t}{N} \hbox{tr}(\dot g) \partial_t u (32)

Thus we see that if u solves the heat equation u_t = \Delta_{g(t)} u, then its lift u^{(N)}: M^{(N)} \to {\Bbb R} is approximately harmonic in the sense that \Delta^{(N)} u^{(N)} = O(1/N) in the region where -t = \tau = O(1) and x is confined to a compact region of space.

Remark 5. The \frac{t}{N} \hbox{tr}(\dot g) \partial_t u 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). \diamond

Now let us see whether Ricci flows t \mapsto (M, g(t)) lift to approximately Ricci-flat manifolds M^{(N)}. We begin by computing the Christoffel symbols (\Gamma^{(N)})^{\gamma}_{\alpha \beta} in local coordinates, where \alpha,\beta,\gamma refer to the N+d combined indices coming from the indices a on M and the indices i on {\Bbb R}^N. We recall the standard formula

\displaystyle \Gamma^{\gamma}_{\alpha \beta} = \frac{1}{2} g^{\gamma \delta} ( \partial_\alpha g_{\beta \delta} + \partial_\beta g_{\alpha \delta} - \partial_\delta g_{\alpha \beta} ) (33)

for the Christoffel symbols of a general Riemannian manifold in local coordinates. Specialising to the metric (25), some computation reveals that

\displaystyle (\Gamma^{(N)})^i_{jk} = 0

\displaystyle (\Gamma^{(N)})^i_{ja} = (\Gamma^{(N)})^i_{aj} = (\Gamma^{(N)})^a_{ij} = 0

\displaystyle (\Gamma^{(N)})^i_{ab} = \frac{y_i}{2N} \dot g_{ab} (34)

\displaystyle (\Gamma^{(N)})^a_{ib} = (\Gamma^{(N)})^a_{bi} = -\frac{y_i}{2N} g^{ac} \dot g_{cb}

\displaystyle (\Gamma^{(N)})^a_{bc} = \Gamma^a_{bc}.

Now the Ricci curvature \hbox{Ric}_{\alpha \beta} can be computed from the Christoffel symbols by the standard formula

\hbox{Ric}_{\alpha \beta} = \partial_\gamma \Gamma^\gamma_{\alpha \beta} - \partial_\beta \Gamma^\gamma_{\alpha \gamma} + \Gamma_{\alpha \beta}^\gamma \Gamma^\mu_{\gamma \mu} - \Gamma^\mu_{\alpha \gamma} \Gamma^\gamma_{\beta \mu}. (35)

If we apply this formula we obtain (after some computation)

\displaystyle \hbox{Ric}^{(N)}_{ij} = \frac{\delta_{ij}}{2N} \hbox{tr}(\dot g) + O( 1 / N^2 )

\displaystyle \hbox{Ric}^{(N)}_{ia} = O( 1 / N ) (36)

\displaystyle \hbox{Ric}^{(N)}_{ab} = \hbox{Ric}_{ab} + \frac{1}{2} \dot g_{ab} + O(1/N).

We thus see that if the original flow t \mapsto (M,g(t)) obeys the Ricci flow equation \dot g = -2 \hbox{Ric}, then the lifted manifold (M^{(N)}, \mu^{(N)}) is nearly Ricci flat in the sense that all components of the Ricci curvature tensor are O(1/N) (in the region t = O(1)). In fact the above estimates show that the Ricci curvature tensor is also O(1/N) in the operator norm sense and O(1/\sqrt{N}) 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 \frac{\delta_{ij}}{2N} \hbox{tr}(\dot g) = - R \delta_{ij} / N term in (36) gives a significant contribution to the trace of the Ricci tensor \hbox{Ric}^{(N)} (i.e. the scalar curvature R^{(N)}), even in the limit N \to \infty. 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 t \mapsto (M,g(t)), define the modified metric \tilde g^{(N)} by the formula

\displaystyle (d\tilde g^{(N)})^2 = dy^2 + \frac{r^2}{N^2} R(t) dr^2 + dg(t)^2 (37)

where of course dr = \sum_{i=1}^N \frac{y_i}{r} dy_i 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

\displaystyle \tilde g^{(N)}_{ij} = \delta_{ij} + \frac{y_i y_j}{N^2} R(t); \tilde g^{(N)}_{ia} = 0; \tilde g^{(N)}_{ab} = g_{ab}. (38)

Exercise 5. Let t \mapsto (M,g(t)) be a smooth ancient Ricci flow on (-\infty,0], and let \tilde g^{(N)} be defined by (37). Show that in the region where y_i = O(1) (so -t = \tau = O(1)) and x ranges in a compact set, the Christoffel symbols (\tilde \Gamma^{(N)})^{\gamma}_{\alpha \beta} take the form

(\tilde \Gamma^{(N)})^i_{jk} = \frac{\delta_{jk}}{N^2} R y_i + O(1/N^3)

(\tilde \Gamma^{(N)})^i_{ja}, (\tilde \Gamma^{(N)})^i_{aj}, (\tilde \Gamma^{(N)})^a_{ij} = O(1/N^2)

(\tilde \Gamma^{(N)})^i_{ab} = \frac{y_i}{2N} \dot g_{ab} + O(1/N^2) (39)

(\tilde \Gamma^{(N)})^a_{ib} = (\tilde \Gamma^{(N)})^a_{bi} = - \frac{y_i}{2N} g^{ac} \dot g_{cb} + O(1/N^2)

(\tilde \Gamma^{(N)})^a_{bc} = \Gamma^a_{bc}.

and the Ricci curvature \widetilde{\hbox{Ric}}^{(N)}_{\alpha \beta} takes the form

\widetilde{\hbox{Ric}}^{(N)}_{ij} = O( 1 / N^2 )

\widetilde{\hbox{Ric}}^{(N)}_{ia} = O( 1 / N ) (40)

\widetilde{\hbox{Ric}}^{(N)}_{ab} = O(1/N).

In particular, \widetilde{\hbox{Ric}}^{(N)} has norm O(1/\sqrt{N}) in the trace (i.e. nuclear) norm (and hence in the Hilbert-Schmidt/Frobenius and operator norms). \diamond

Exercise 6. Let the assumptions and notation be as in Exercise 5, let u: (-\infty,0] \times M \to {\Bbb R} be a smooth function, and let u^{(N)} be as in (31). Show that the Laplacian \tilde \Delta^{(N)} associated to \tilde g^{(N)} obeys a similar formula to (32), but with the \frac{r}{2N} \hbox{tr}(\dot g) \partial_r u^{(N)} term replaced by terms which are O(1/N^2) when t, x are bounded. \diamond

— Perelman’s reduced length and reduced volume —

In the previous discussion, we have converted a Ricci flow t \mapsto (M,g(t)) to a Riemannian manifold (M^{(N)}, \tilde g^{(N)}) 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 N \to \infty), thus the Bishop-Gromov reduced volume r_0^{-N-d} B_{\tilde g^{(N)}}((0,x_0),r_0) should heuristically be non-increasing in r_0, where we fix a spatial origin x_0 \in M.

In order to exploit the above heuristic, we first need to understand the distance function on (\tilde M, g(t)). Let (y_1,x_1) = (r_1\omega_1, x_1) be a point in \tilde M = {\Bbb R}^N \times M, and consider a length-minimising geodesic \gamma^{(N)}: [0,\tau_1] \to \tilde M from (0,x_0) to (r_1\omega_1, x_1), where we have normalised the length \tau_1 of the parameter interval by the formula \tau_1 = r_1^2/2N.

Observe that the metric (37) can be rewritten in polar coordinates (after substituting -t = \tau = r^2/2N) as

\displaystyle (d\tilde g^{(N)})^2 = (\frac{N}{2\tau} + R) d\tau^2 + 2N\tau d\omega^2 + dg(-\tau)^2 (41)

(which is essentially the first formula in Section 6 of Perelman’s paper). Note that the angular variable \omega only influences the second term in this metric and not the other two. Because of this, one sees that the geodesic \gamma^{(N)} must keep \omega constant in order to be length-minimising (i.e. \omega = \omega_1 for the duration of the geodesic). Turning next to the \tau variable, we then see that for N large enough, the geodesic \gamma^{(N)} should increase \tau continuously from 0 to \tau_1 (as the \frac{N}{2\tau} term in (41) will severely penalise any backtracking. After a reparameterisation we may in fact assume that \tau increases at constant speed, thus we have

\displaystyle \gamma^{(N)}(\tau) = ( \sqrt{2N\tau} \omega_1, \gamma(\tau) ) (42)

for some path \gamma: [0,\tau_1] \to M from x_0 to x_1. Using (41), the length of this geodesic is

\displaystyle \int_0^{\tau_1} \sqrt{ \frac{N}{2\tau} + R + |\gamma'(\tau) |_{g(-\tau)}^2}\ d\tau (43)

which by Taylor expansion is equal to

\displaystyle \sqrt{2N\tau_1} + \frac{1}{\sqrt{2N}} {\mathcal L}(\gamma) + O(N^{-3/2}) (44)

where the {\mathcal L}-length of \gamma is defined as

{\mathcal L}(\gamma) := \int_0^{\tau_1} \sqrt{\tau} ( R + |\gamma'(\tau)|_{g(-\tau)}^2 )\ d\tau (45)

Note that this quantity is independent of N. Thus, heuristically, geodesics in {\mathcal M} from (0,x_0) to (r_1 \omega_1,x_1) should (approximately) minimise the {\mathcal L}-length. If we define L_{(0,x_0)}(-\tau_1,x_1) to be the infimum of {\mathcal L}(\gamma) over all paths \gamma: [0,\tau_1] \to M from x_0 to x_1, we thus obtain the heuristic approximation

\displaystyle d_{\tilde g^{(N)}}( (0,x_0), (r_1 \omega_1, x_1) ) = \sqrt{2N\tau_1} + \frac{1}{\sqrt{2N}} L_{(0,x_0)}(-\tau_1,x_1). (46)

Exercise 7. When M is the Euclidean space {\Bbb R}^d (with the trivial Ricci flow, of course), show that L_{(0,x_0)}(-\tau_1,x_1) = |x_1-x_0|^2 / 2 \sqrt{\tau_1}, and the minimiser is given by \gamma(\tau) = x_0 + \sqrt{\frac{\tau}{\tau_1}} (x_1-x_0). \diamond

From (46) we see that the ball in (M^{(N)}, \tilde g^{(N)}) of radius r_0 = \sqrt{2N \tau_0} centred at (0,x_0) (where, as always, we are in the regime \tau_0 = O(1), so r_0^2 = O(N)) should heuristically take the form

\{ (r_1 \omega_1, x_1): L_{(0,x_0)}(-\tau_1,x_1) \leq 2N (\sqrt{\tau_0}-\sqrt{\tau_1}) \}. (47)

If we make the plausible assumption that L_{(0,x_0)}(-\tau,x) varies smoothly in \tau, then (47) is heuristically close (when N is large) to

\{ (r_1 \omega_1, x_1): L_{(0,x_0)}(-\tau_0,x_1) \leq 2N (\sqrt{\tau_0}-\sqrt{\tau_1}) \} (48)

or equivalently

\{ (r_1 \omega_1, x_1): r_1 \leq r_0 - \frac{1}{\sqrt{2N}} L_{(0,x_0)}(-\tau_0,x_1) \}. (49)

Now, the volume measure of (37) is of the form (1 + O(1/N)) dy d\mu(t), and so the volume of (49) is approximately

\displaystyle C_N \int_M \int_0^{r_0 - \frac{1}{\sqrt{2N}} L_{(0,x_0)}(-\tau_0,x_1)} r_1^{N-1}\ dr_1 d\mu(t_1)(x_1). (50)

(Note there is a slight abuse of notation since t_1 depends on r_1, 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

\displaystyle \frac{1}{N} C_N \int_M (r_0 - \frac{1}{\sqrt{2N}} L_{(0,x_0)}(-\tau_0,x))^N d\mu(t_0)(x). (51)

We can Taylor expand this to be approximately

\displaystyle \frac{1}{N} C_N r_0^N \int_M \exp(- l_{(0,x_0)}(-\tau_0,x) )\ d\mu(t_0)(x) (52)

where the Perelman reduced length l_{(0,x_0)}(-\tau_0,x) is defined as

\displaystyle l_{(0,x_0)}(-\tau,x) := \frac{L_{(0,x_0)}(-\tau,x)}{2\sqrt{\tau}} = \frac{\sqrt{2N} L_{(0,x_0)}(-\tau,x)}{2r_0} (53)

Example 1. Continuing the Euclidean example of Exercise 7, we have l_{(0,x_0)}(-\tau,x) = |x-x_0|^2/4\tau, which is the familiar exponent in the fundamental solution (15). This is, of course, not a coincidence. \diamond

From (52) we thus heuristically conclude that the Bishop-Gromov reduced volume of (M^{(N)}, \tilde g^{(N)}) at (0,x_0) and at radius r_0 = \sqrt{2N\tau_0} is approximately equal to a constant multiple of \tilde V_{(0,x_0)}( -\tau ), where the Perelman reduced volume \tilde V_{(0,x_0)}( -\tau ) is defined as

\displaystyle \tilde V_{(0,x_0)}( -\tau ) := \int_M \tau^{-d/2} \exp( - l_{(0,x_0)}( -\tau,x) )\ d\mu(-\tau)(x). (54)

Example 2. Again continuing the Euclidean example, the reduced volume in Euclidean space (with the trivial Ricci flow) is always (4\pi)^{d/2}. \diamond

Formally applying Corollary 1, we are thus led to

Conjecture 1. (Monotonicity of Perelman reduced volume) Let t \mapsto (M,g(t)) be a Ricci flow on {}[-T,0], and let x_0 \in M_0. Then the quantity \tilde V_{(0,x_0)}( -\tau ) for 0 < \tau \leq T is monotone non-increasing in \tau.

Remark 6. Note here we are not taking the Ricci flow to be ancient; this would correspond to the manifold M^{(N)} being replaced by an incomplete manifold, of radius about \sqrt{2NT}. However, because of the restriction \tau \leq T, 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. \diamond

Remark 7. At an crude heuristic level, the Perelman reduced volume \tilde V_{(0,x_0)}( -\tau ) is roughly like \hbox{Vol}_{g(-\tau)}( -\tau, O(\sqrt{\tau}) ) / \tau^{d/2} (since, in view of Exercise 7, we expect l_{(0,x_0)}(-\tau,x) to behave like d(x_0,x)^2/\tau, 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. \diamond

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 {\mathcal L}-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. \diamond