Planet Musings

May 09, 2008

Cosmic Variance — Props to Mitt Romney

Reflecting on his earlier speech on faith at the height of his campaign, Mitt decides to stand up for atheists:

But upon reflection, I realized that while I could defend their absence from my address, I had missed an opportunity…an opportunity to clearly assert that non-believers have just as great a stake as believers in defending religious liberty.

If a society takes it upon itself to prescribe and proscribe certain streams of belief — to prohibit certain less-favored strains of conscience — it may be the non-believer who is among the first to be condemned. A coercive monopoly of belief threatens everyone, whether we are talking about those who search the philosophies of men or follow the words of God.

We are all in this together. Religious liberty and liberality of thought flow from the common conviction that it is freedom, not coercion, that exalts the individual just as it raises up the nation.

(from a speech at the “Beckett Fund for Religious Liberty’s Canterbury dinner”). He loses a lot of ground with me on the rest of the speech, where he elaborates on his earlier claim that “freedom requires religion” and argues that without religion keeping us all well behaved, the US would have decended into anarchy or facism. All the same, it’s nice to see someone tied so closely with both politics and faith demonstrating understanding of why atheists get a bit squicked out with the notion of theocracies.

by Julianne at May 09, 2008 06:57 PM

Steinn Sigurðsson — flu like illness on Canadian train

Passenger train halted and quarantined in northern Ontario

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Cosmic Variance — My New Favorite Conference Title

I just received the following workshop announcement:

11th Birmingham-Nottingham Extragalactic Workshop - 1st Announcement
“Semi-analytic models - are we kidding ourselves?“

Refeshingly honest conference title aside, this is a terrific topic for a workshop. Semi-analytic galaxy formation models are extremely useful tools, which consist of (1) an underlying prescription for the growth of dark matter halos and (2) a set of knobs for grafting complicated baryonic physics onto those halos. The first step is well-understood analytically, and has been reliably calibrated with N-body simulations. The second step, however, contains a lot of crafty juju (how much gas winds up inside the dark matter halos? At what rate does that gas cool? How and when does that gas convert into stars? How does the formation of stars and the subsequent supernovae affect the surrounding gas? How do mergers between dark matter halos change the spatial distribution of the stars and the temperature structure of the gas?). The developers of these models are smart folks, and make reasonably well-informed assumptions about all of these baryonic processes, leading to results that are decent matches to the ensemble properties of the galaxy populations. (Note that I didn’t say “predict results” — these models are ex post facto in most usages.) However, just because the models can be tuned to produce a rough statistical match to observations in no way means that the input assumptions are correct or unique descriptions of what actually happens. Moreover, there are serious discrepancies between the models and the observed properties of very low mass galaxies — when the models are tuned to match the properties of relatively massive galaxies, they predict that the low mass galaxies are red and gas poor, whereas the observations say they’re blue and gas rich. It’s great that the community is looking at these issues head on, given the usefulness the semi-analytic galaxy models.

by Julianne at May 09, 2008 06:00 PM

Dave Bacon — Algorithmic Steampunk?

From the annals of "is that really the word you wanted?" from a New York Times article on steampunk:

"There seems to be this sort of perfect storm of interest in steampunk right now," Mr. von Slatt said. "If you go to Google Trends and track the number of times it is mentioned, the curve is almost algorithmic from a year and a half ago." (At this writing, Google cites 1.9 million references.)

Certainly I can interpret this as saying that the trend has a curve which can be generated by an algorirthm, but I'm guessing Mr. von Slatt meant something else, considering that the curve which is always zero is also algorithmic. Read the comments on this post...

Chad Orzel — Interdisciplinarity

Timothy Burke has some interesting thoughts about the College of the Atlantic, which represents a real effort to build interdisciplinarity on an institutional level. "Interdisciplinary" is the buzzword of the moment in large swathes of academia, and the College of the Atlantic, which doesn't have departments and works very hard to make connections between disciplines, is sort of the apotheosis of the interdisciplinary movement.

Toward the end of his post, Burke relates a story from earlier in his career:

When I was briefly at Emory at the start of my career, I was in a workshop on interdisciplinarity. After many of the junior people there duly celebrated interdisciplinary study, a very wise, interesting senior professor, a classicist, stirred himself. "You guys," he said, "don't have a beef with disciplines. You have a beef with departments. Everybody's interdisciplinary in some respect in their scholarship. It's departments that cause the problem by raising the barriers to interaction and discussion".

To some degree, I agree with this-- a lot of the problem with interdisciplinary work has to do with the structure of academia, not the content of the disciplines. But we shouldn't overlook the fact that there are very real differences in the ways that scholars in different disciplines go about their business. The most extreme example, and the one that causes the most angst when discussing merit evaluations, has to do with book publishing-- scholars in the humanities and social sciences are expected to produce scholarly monographs, but that's almost unheard of in the natural sciences and engineering. This makes it very difficult to construct a system that appropriately rewards both humanists and scientists for their scholarly production.

But there are significant differences even within the sciences:

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Steinn Sigurðsson — teh science frame

Nima Arkani-Hamed front pages cnn.com

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Dave Bacon — Turok New PI Director

Of interest to quantum computeristas: Cosmologist Niel Turok has been named the new director of the Perimeter Institute. Onward and upward!

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Terence Tao — 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume

Having completed a heuristic derivation of the monotonicity of Perelman reduced volume (Conjecture 1 from the previous lecture), we now turn to a rigorous proof. Whereas in the previous lecture we derived this monotonicity by converting a parabolic spacetime to a high-dimensional Riemannian manifold, and then formally applying tools such as the Bishop-Gromov inequality to that setting, our approach here shall take the opposite tack, finding parabolic analogues of the proof of the elliptic Bishop-Gromov inequality, in particular obtaining analogues of the classical first and second variation formulae for geodesics, in which the notion of length is replaced by the notion of ${\mathcal L}$ -length introduced in the previous lecture.

The material here is primarily based on Perelman’s first paper and Müller’s book, but detailed treatments also appear in the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.

– Reduction to a pointwise inequality –

Recall that the Bishop-Gromov inequality (Corollary 1 from the previous lecture) states (among other things) that if a d-dimensional complete Riemannian manifold (M,g) is Ricci-flat (or more generally, has non-negative Ricci curvature), and $x_0$ is any point in M, then the Bishop-Gromov reduced volume $\hbox{Vol}(B(x_0,r))/r^d$ is a non-decreasing function of r. In fact one can obtain the slightly sharper result that $\hbox{Area}(S(x_0,r))/r^{d-1}$ is a non-decreasing function of r, where $S(x_0,r)$ is the sphere of radius r centred at $x_0$ .

From the basic formula ${\mathcal L}_{\partial r} d\mu = (\Delta r)\ d\mu$ (equation (1) from the previous lecture) and the Gauss lemma, one readily obtains the identity

$\displaystyle \frac{d}{dr} \hbox{Area}(S(x_0,r)) = \int_{S(x_0,r)} \Delta r\ dS$ (1)

where $dS$ is the area element. The monotonicity of $\hbox{Area}(S(x_0,r))/r^{d-1}$ then follows (formally, at least) from the pointwise inequality

$\displaystyle \Delta r \leq \frac{d-1}{r}$ (2)

which we will derive shortly (at least for the portion of the manifold inside the cut locus) as a consequence of the first and second variation formulae for geodesics. (In the previous lecture, the inequality (2) was derived from a transport inequality for $\Delta r$ , but we will take a slightly different tack here.) Observe that (2) is an equality when (M,g) is a Euclidean space ${\Bbb R}^d$ .

It turns out that the monotonicity of Perelman reduced volume for Ricci flows can similarly be reduced to a pointwise inequality, in which the Laplacian $\Delta$ is replaced by a heat operator, and the radial variable r is replaced by the Perelman reduced length. More precisely, given an ancient Ricci flow $t \mapsto (M,g(t))$ for $t \in (-\infty,0]$ , a time $-\tau$ , and two points $x_0, x \in M$ , recall that the reduced length $l_{(0,x_0)}( -\tau,x)$ is defined as

$\displaystyle l_{(0,x_0)}( -\tau,x) := \frac{1}{2\sqrt{\tau}} \inf_\gamma {\mathcal L}(\gamma)$ (3)

where the ${\mathcal L}$ -length ${\mathcal L}(\gamma)$ of a curve $\gamma: [0,\tau_1] \to M$ from $x_0$ to $x_1$ is defined as

$\displaystyle {\mathcal L}(\gamma) = \int_0^{\tau_1} \sqrt{\tau} (R + |X|_{g(-\tau)}^2)\ d\tau$ , (4)

where we adopt the shorthand $X := \partial_{\tau} \gamma$ , and that Conjecture 1 from the previous lecture asserts that the Perelman reduced volume

$\displaystyle \tilde V_{(0,x_0)}(-\tau) = \int_M \tau^{-d/2} \exp( - l_{(0,x_0)}(-\tau,x) )\ d\mu_{g(-\tau)}(x)$ (5)

is non-decreasing in $\tau$ for Ricci flows. If we differentiate (5) in $\tau$ , using the variation formula $\frac{d}{d\tau} d\mu = R\ d\mu$ , we easily verify that the monotonicity of (5) will follow (assuming $l_{(0,x_0)}$ is sufficiently smooth, and that either M is compact, or $l_{(0,x_0)}$ grows sufficiently quickly at infinity) from the pointwise inequality

$\displaystyle \partial_{\tau} l_{(0,x_0)} - \Delta_{g(-\tau)} l_{(0,x_0)} + |\nabla l|_{g(-\tau)}^2 - R + \frac{d}{2\tau} \geq 0$ (6)

which should be viewed as a parabolic analogue to (2).

Exercise 1. Verify that (6) is an equality in the case of the (trivial) Ricci flow on Euclidean space, using Example 1 from the previous lecture. (This is of course consistent with Example 2 from that lecture.) $\diamond$

Exercise 2. Show that (6) is equivalent to the assertion that the function $v(-\tau,x) := (4\pi \tau)^{-d/2} \exp(-l_{(0,x_0)}(-\tau,x))$ is a subsolution of the adjoint heat equation, or more precisely that $\partial_t v - \Delta v + Rv \leq 0$ . Note that this fact implies the monotonicity of Perelman reduced volume (cf. Exercise 2 from Lecture 8). [It seems that the elliptic analogue of this fact is the assertion that the Newton-type potential $1/r^{d-2}$ is subharmonic away from the origin for Ricci flat manifolds of dimension three or larger , which is a claim which is easily seen to be equivalent to (2) thanks to the Gauss lemma.] $\diamond$

So to prove monotonicity of the Perelman reduced volume, the main task will be to establish the pointwise inequality (6). (There are some additional technical issues, mainly concerning the parabolic counterpart of the cut locus, which we will also have to address, but we will work formally for now, and deal with these analytical matters later.)

We will perform a minor simplification: by using the rescaling symmetry $g(t,x) \mapsto \lambda^2 g(\frac{t}{\lambda^2})$ (and noting the unsurprising fact that (6) is dimensionally consistent) we can normalise $\tau_1 = 1$ .

– First and second variation formulae for ${\mathcal L}$ -geodesics –

To establish (6), we of course need some variation formulae that compute the first and second derivatives of the reduced length function $l_{(0,x_0)}$ . To motivate these formulae, let us first recall the more classical variation formulae that give the first and second derivatives of the metric function $d(x_0,x)$ on a Riemannian manifold (M,g), which in particular can be used to derive (2) when the Ricci curvature is non-negative.

We recall that the distance $d(x_0,x)$ can be defined by the energy-minimisation formula

$\displaystyle \frac{1}{2} d(x_0,x)^2 = \inf_\gamma E(\gamma)$ (7)

where $\gamma: [0,1] \to M$ ranges over all $C^1$ curves from $x_0$ to x, where the Dirichlet energy $E(\gamma)$ of the curve is given by the formula

$\displaystyle E(\gamma) = \frac{1}{2} \int_0^1 |X|_g^2\ dt$ (8)

where we write $X := \partial_t \gamma$ . It is known that this infimum is always attained by some geodesic $\gamma$ ; we shall assume this implicitly in the computations which follow.

Now suppose that we deform such a curve $\gamma$ with respect to a real parameter $s \in (-\varepsilon,\varepsilon)$ , thus $\gamma: (s,t) \mapsto \gamma(s,t)$ is now a function on the two-dimensional parameter space $(\varepsilon,\varepsilon) \times [0,1]$ . The first variation here can be computed as

$\displaystyle \frac{d}{ds} E(\gamma) = \int_0^1 g( \nabla_X X, X )\ dt$ (9)

where $\nabla_X$ is the pullback of the Levi-Civita connection on M with respect to $\gamma$ applied in the direction $\partial_t$ ; here we of course use that g is parallel with respect to this connection. The torsion-free nature of this connection gives us the identity

$\nabla_Y X = \nabla_X Y$ (10)

where $Y = \partial_s \gamma$ is the infinitesimal variation, and $\nabla_Y$ is the pullback of the Levi-Civita connection applied in the direction $\partial_s$ (cf. Exercise 5 from Lecture 6). An integration by parts (again using the parallel nature of g) then gives the first variation formula

$\displaystyle \frac{d}{ds} E(\gamma) = g( Y, X )|_{t=0}^1 - \int_0^1 g( Y, \nabla_X X )\ dt$ . (11)

If we fix the endpoints of $\gamma$ to be $\gamma(s,0) = x_0$ and $\gamma(s,1)=x_1$ , then the first term on the right-hand side of (11) vanishes. If we consider arbitrary infinitesimal variations $Y$ of $\gamma$ with fixed endpoints, we thus conclude that in order to be a minimiser for (7), that $\gamma$ must obey the geodesic flow equation

$\nabla_X X = 0$ . (12)

One consequence of this is that the speed $|X|_g$ of such a minimiser must be constant, and from (7) we then conclude

$|X|_g = d(x_0,x)$ . (13)

If we then vary a geodesic $\gamma(0,\cdot)$ with the initial endpoint $\gamma(s,0)$ fixed at $x_0$ and the final endpoint $\gamma(s,1) = x(s)$ variable, the variation formula (11) gives

$\displaystyle \frac{d}{ds} E(\gamma) = g( x'(s), X(s,1) )$ (14)

which, if we insert this back into (7) and use (13), gives

$\displaystyle \frac{d}{ds} d(x_0,x) \leq g( x'(s), X(s,1) / |X(s,1)|_g )$ (13)

which is a (one-sided) version of the Gauss lemma. If one is inside the cut locus, then the metric function is smooth, and one can then replace the inequality with an equality by considering variations both forwards and backwards in the s variable, recovering the full Gauss lemma. In particular, we conclude in this case that $\nabla d(x_0,x)$ is a unit vector.

Now we consider the second variation $\frac{d^2}{ds^2} E(\gamma)$ of the energy, when $\gamma$ is already a geodesic. For simplicity we assume that $\gamma$ evolves geodesically in the s direction, thus

$\nabla_Y Y = 0$ . (14)

[Actually, since $\gamma$ is already a geodesic and thus is stationary with respect to perturbations that respect the endpoints, the values of $\nabla_Y Y$ away from endpoints - which represents a second-order perturbation respecting the endpoints - will have no ultimate effect on the second variation of $E(\gamma)$ . Nevertheless it is convenient to assume (14) to avoid a few routine additional calculations.]

Differentiating (9) once more we obtain

$\displaystyle \frac{d^2}{ds^2} E(\gamma) = \int_0^1 g( \nabla_Y \nabla_Y X, X ) + |\nabla_Y X|_g^2 \ dt$ . (15)

Using (10), (14), and the definition of curvature, we have

$\nabla_Y \nabla_Y X = \nabla_Y \nabla_X Y$

$= \nabla_X \nabla_Y Y + \hbox{Riem}( Y, X ) Y$ (16)

$= -\hbox{Riem}( X, Y ) Y$

and thus (by one further application of (10))

$\displaystyle \frac{d^2}{ds^2} E(\gamma) = \int_0^1 |\nabla_X Y|_g^2 - g(\hbox{Riem}(X, Y) Y, X)\ dt$ . (17)

Now let us fix the initial endpoint $\gamma(s,0) = x_0$ and let the other endpoint $\gamma(s,1) = x(s)$ vary, thus $\partial_s \gamma$ equals 0 at time t=0 and equals $x'(s)$ at time t=1. From Cauchy-Schwarz we conclude

$\int_0^1 |\nabla_X Y|_g^2\ dt \leq \int_0^1 (\partial_t |Y|_g)^2\ dt \leq (\int_0^1 \partial_t |Y|_g\ dt)^2 = |x'(s)|^2$ . (18)

Actually, we can attain equality here by choosing the vector field $Y$ appropriately:

Exercise 3. If we set $Y := t v$ , where v is the parallel transport of x’(s) along X, or more precisely the vector field that solves the ODE

$\nabla_X v = 0; v(s,1) = x'(s)$ (19)

show that all the inequalities in (18) are obeyed with equality. $\diamond$

For such a vector field, we conclude that

$\displaystyle \frac{d^2}{ds^2} E(\gamma) = |x'(s)|^2 - \int_0^1 - t^2 g(\hbox{Riem}(X, v) v, X)\ dt$ . (20)

From this formula (and the first variation formula) we conclude that

$\displaystyle \frac{d^2}{ds^2} \frac{1}{2} d(x_0,x)^2 \leq |x'(s)|^2 - \int_0^1 t^2 g(\hbox{Riem}(X, v) v, vX\ dt$ . (21)

Now let $x'(0)$ vary over an orthonormal basis of the tangent space of x(0); by (19) we see that v determines an orthonormal frame for s=0 and $0 \leq t \leq 1$ . Summing (21) over this basis (and using the formula for the Laplacian in normal coordinates) we conclude that

$\displaystyle \Delta \frac{1}{2} d(x_0,x)^2 \leq d - \int_0^1 t^2 \hbox{Ric}(X,X)\ dt$ . (22)

In particular, for manifolds of non-negative Ricci curvature we have

$\displaystyle \Delta \frac{1}{2} d(x_0,x)^2 \leq d$ (23)

from which (2) easily follows from the Gauss lemma. (Observe that (23) is obeyed with equality in the Euclidean case.)

Now we develop analogous variational formulae for ${\mathcal L}$ -length (and reduced length) on a Ricci flow. We shall work formally for now, assuming that all infima are actually attained and that all quantities are as smooth as necessary for the analysis that follows to work; we then discuss later how to justify all of these assumptions. As mentioned earlier, we normalise $\tau_1 = 1$ .

Let us take a path $\gamma: [0,1] \to M$ and vary it with respect to some additional parameter s as before. Differentiating (4), we obtain

$\displaystyle \frac{d}{ds} {\mathcal L}(\gamma) = \int_0^1 \sqrt{\tau} ( \nabla_Y R + 2 g( X, \nabla_Y X ))\ d\tau$ (24)

where $X := \partial_\tau \gamma$ and $Y := \partial_s \gamma$ . On the other hand, if we have a Ricci flow $\partial_\tau g = 2 \hbox{Ric}$ , we see that

$\partial_\tau g( X, Y ) = g( \nabla_X X, Y ) + g( X, \nabla_X Y ) + 2 \hbox{Ric}( X, Y )$ ; (25)

placing this into (24) and using the fundamental theorem of calculus, we can express the right-hand side of (24) as

$2 g( X, Y )(\tau_1) - 2 \int_0^1 \sqrt{\tau} g( Y, G(X) )\ d\tau$ (26)

where G(X) is the vector field

$G := \nabla_X X - \frac{1}{2} \nabla R + \frac{1}{2\tau} X + 2 \hbox{Ric}(X,\cdot)$ . (27)

Note that G does not depend on Y. From this we see that in order for $\gamma$ to be a minimiser of ${\mathcal L}(\gamma)$ with the endpoints fixed, we must have G(X)=0, which is the parabolic analogue of the geodesic flow equation (14).

Example 1. In the case of the trivial Euclidean flow, the minimal ${\mathcal L}$ -path from $(0,x_0)$ to $(-1,x_1)$ takes the form $\gamma(\tau) = x_0 + v \sqrt{\tau}$ where $v := x_1-x_0$ , in which case $X = \frac{v}{2\sqrt{\tau}}$ . It is not hard to verify that G=0 in this case. $\diamond$ .

Arguing as in the elliptic case, we conclude (assuming the existence of a unique minimiser, and the local smoothness of reduced length) the first variation formula

$\partial_s l_{(0,x_0)}(-1,x_1) = g( X, \partial_s x_1)(1)$ (28)

or equivalently

$\nabla l_{(0,x_0)}(-1,x_1) = X(1)$ . (29)

Example 2. Continuing Example 1, note that $l_{(0,x_0)}(-1,x_1) = |x_1-x_0|^2/4$ and $\partial_\tau \gamma = (x_1-x_0)/2$ , which is of course consistent with (28). $\diamond$

Having computed the spatial derivative of the reduced length, we turn to the time derivative. The simplest way to compute this is to observe that any partial segment of an ${\mathcal L}$ -minimising path must again be a ${\mathcal L}$ -minimising path. From (4) and the fundamental theorem of calculus we have

$\displaystyle \frac{d}{d\tau_1} {\mathcal L}(\gamma)|_{\tau_1 = 1} = R + |X|_g^2$ (30)

where we vary $\gamma$ in $\tau_1$ by truncation; by (3) and the above discussion we conclude

$\displaystyle \frac{d}{d\tau_1} ( 2\sqrt{\tau_1} l_{(0,x_0)}(-\tau_1,x_1) )|_{\tau_1=1} = (R + |X|_g^2)$ (31)

where $(\tau_1,x_1)$ varies along $\gamma$ (in particular, $\partial_{\tau_1} x_1 = X$ ). Applying the product and chain rules, we can expand the left-hand side of (31) as

$l_{(0,x_0)}(-1,x_1) + 2 \partial_{\tau_1}l_{(0,x_0)}(-\tau_1,x_1)|_{\tau_1=-1} + 2 g( \nabla l_{(0,x_0)}(-1,x_1), X )$ ; (32)

using (29), we conclude that

$\partial_{\tau_1} l_{(0,x_0)}(-\tau_1,x_1)|_{\tau_1=1} = \frac{1}{2} (R + |X|_g^2) - \frac{1}{2} l_{(0,x_0)}(-1,x_1) - |X|_g^2$ . (33)

Now we turn to the second spatial variation of the reduced length. Let $\gamma$ be a ${\mathcal L}$ -minimiser, so that G=0. Differentiating (24) again, we obtain

$\displaystyle \frac{d^2}{ds^2} {\mathcal L}(\gamma) = \int_0^{1} \sqrt{\tau} ( \nabla_Y \nabla_Y R + 2 |\nabla_Y X|^2 + 2 g( X, \nabla_Y \nabla_Y X ))\ d\tau$ . (34)

As in the elliptic case, it is convenient to assume that we have a geodesic variation (14). In that case, we again have (16), and we also have $\nabla_Y \nabla_Y R = \hbox{Hess}(R)(Y,Y)$ . Using (10), we thus express (34) as

$\int_0^{1} \sqrt{\tau} ( \hbox{Hess}(R)(Y,Y) + 2 |\nabla_X Y|^2 - 2 g(\hbox{Riem}(X,Y) Y, X))\ d\tau$ . (35)

As before, we optimise this in Y. Because the metric g now changes in time by Ricci flow, one has to modify the prescription in Exercise 3 slightly. More precisely, we now set $Y := \sqrt{\tau} v$ , where v solves the following variant of (19),

$\nabla_X v = - \hbox{Ric}(v,\cdot); v(s,1) = x'(s)$ . (36)

The point of doing this is that the ODE is orthogonal; the length of v is preserved along X, as is the inner product between any two such v’s (cf. equation (15) from Lecture 3). A brief computation then shows that

$\displaystyle \nabla_X Y = \frac{1}{2\sqrt{\tau}} v - \sqrt{\tau} \hbox{Ric}(v,\cdot)$ (37)

and hence

$\displaystyle |\nabla_X Y|_g^2 = \frac{1}{4\tau} |x'(s)|_g^2 + \tau |\hbox{Ric}(v,\cdot)|^2 - \hbox{Ric}(v,v)$ . (38)

Putting all of this into (35), we now see that the second variation (34) is equal to

$\displaystyle \int_0^1 \tau^{3/2} \hbox{Hess}(R)(v,v) + \frac{1}{2\tau^{1/2}} |x'(s)|_g^2 + 2\tau^{3/2} |\hbox{Ric}(v,\cdot)|^2 - 2 \tau^{1/2} \hbox{Ric}(v,v) - 2\tau^{3/2} g(\hbox{Riem}(X,v) v, X)\ d\tau$ . (39)

We now let $x'(0)$ range over an orthonormal basis of $x(0)$ , which leads to v being an orthonormal frame at every point (0,t). Summing over (39) and also using (3), we conclude that

$\displaystyle \Delta l_{(0,x_0)}(-1,x_1) \leq \int_0^{1} \frac{\tau^{3/2}}{2} \Delta R + \frac{d}{4\tau^{1/2}} + \tau^{3/2}|\hbox{Ric}|_g^2 - \tau^{1/2} R - \tau^{3/2} \hbox{Ric}(X,X)\ d\tau$ . (40)

Now we simplify the right-hand side of (40). The second term is of course elementary:

$\displaystyle \int_0^1 \frac{d}{4\tau^{1/2}}\ d\tau = \frac{d}{2}$ (41)

and this is consistent with the Euclidean case (in which $\Delta l_{(0,x_0)}$ is exactly $\frac{d}{2}$ when $\tau_1=1$ , and all curvature terms vanish). To simplify the remaining terms, we recall the variation formula

$-\partial_\tau R = \Delta R + 2 |\hbox{Ric}|_g^2$ (42)

for the scalar curvature (equation (31) of Lecture 1); by the chain rule, we thu have the total derivative formula

$\frac{d}{d\tau} R = -\Delta R - 2 |\hbox{Ric}|_g^2 + \nabla_X R$ . (43)

Inserting (41), (43) into (40) and integrating by parts, we express the right-hand side of (40) as

$\displaystyle \frac{d}{2} - \frac{1}{2} R + \int_0^{1} \frac{\tau^{3/2}}{2} \nabla_X R - \frac{\tau^{1/2}}{4} R - \tau^{3/2} \hbox{Ric}(X,X)\ d\tau$ . (44)

To simplify this further, recall that the quantity G defined in (27) vanishes. This (and the fact that g evolves by Ricci flow $\partial_\tau g = 2 \hbox{Ric}$ ) allows one to compute the variation of $\tau |X|_g^2$ :

$\partial_\tau (\tau |X|_g^2) = \tau \partial_X R - 2 \tau \hbox{Ric}(X,X)$ . (45)

Inserting this into (44) and integrating by parts, one can rewrite (44) as

$\displaystyle \frac{d}{2} - \frac{1}{2} R + \frac{1}{2} |X|_g^2 - \frac{1}{4} \int_0^1 \sqrt{\tau} (R + |X|^2)\ d\tau$ (46)

and so by (3) we obtain the inequality

$\Delta l_{(0,x_0)}(-1,x_1) \leq \frac{d}{2} - \frac{1}{2} R + \frac{1}{2} |X|_g^2 - \frac{1}{2} l_{(0,x_0)}(-1,x_1)$ . (47)

Combining (29), (33), and (47) we obtain (6) as desired.

– Analytical issues –

We now discuss in broad terms the analytical issues that one must address in order to make the above arguments rigorous. We first review the classical elliptic theory (i.e. the theory of geodesics in a Riemannian manifold) before turning to Perelman’s parabolic theory of ${\mathcal L}$ -geodesics in a flow of Riemannian metrics.

In a complete Riemannian manifold, a geodesic $\gamma: [0,1] \to M$ from a fixed point $\gamma(0) = x_0$ to some other point $\gamma(1)=x_1$ has a well-defined initial velocity vector $\gamma'(0) = X(0)$ , and conversely each initial velocity vector $v = X(0)\in T_{x_0} M$ determines a unique geodesic with an endpoint $x_1 = \exp_{x_0}(v)$ , thus defining the exponential map based at $x_0$ . One can show (from standard ODE theory) that this exponential map is smooth (with the derivative of this map controlled by Jacobi fields). Also, if M is connected, then any two points can be joined by a geodesic, and the exponential map is onto. However, there can be vectors v for which this map degenerates (i.e. its derivative ceases to be invertible) - these correspond to the conjugate points of $x_0$ in M.

Define the injectivity region of $x_0$ to be the set of all $x_1$ for which there is a unique minimising geodesic from $x_0$ to $x_1$ , and that the exponential map is not degenerate along this geodesic (in particular, $x_0$ and $x_1$ are not conjugate points). An analysis of Jacobi fields reveals that the injectivity region is open, that the distance function is smooth in this region (except at the origin), and that all the computations given above for the distance function can be justified. So it remains to understand what happens on the complement of the injectivity region, known as the cut locus. Points on the cut locus are either conjugate points to $x_0$ , or are else places where minimising geodesics are not unique, which (by a variant of the Gauss lemma) forces the distance function to be non-differentiable at these points. The former type of points form a set of measure zero, thanks to Sard’s theorem, whereas the latter set of points also form a set of measure zero, thanks to Radamacher’s differentiation theorem and the Lipschitz nature of the distance function (i.e. the triangle inequality). Thus the injectivity region has full measure. While this does mean that pointwise inequalities such as (2) now hold almost everywhere, this is unfortunately not quite enough to ensure that (2) holds in the sense of distributions, which is what one really needs in order to fully justify results such as the Bishop-Gromov inequality. (Indeed, by considering simple examples such as the unit circle, we see that the distribution $\Delta r$ can in fact contain some negative singular measures, although one should note that this does not actually contradict (2) due to the favourable sign of these singular components.) Fortunately, one can address this technical issue by constructing barrier functions to the radius function r at every point $x_1$ , i.e. $C^2$ functions $u=u_{\varepsilon}$ for each $\varepsilon > 0$ which upper bound r near $x_1$ (and match r exactly at $x_1$ , and which obeys the inequality (2) at $x_1$ up to a loss of $\varepsilon$ . Such functions can be constructed at any $x_1$ , even those in the cut locus, by perturbing the origin $x_0$ by an epsilon, and then one can use these barrier functions to justify (2) in the sense of distributions. (I believe that these arguments to control the distance function outside of the injectivity region originate with a paper of Calabi.) From this one can rigorously justify the Bishop-Gromov inequality for all radii, even those exceeding the radius of injectivity.

Analogues of the above assertions hold for the monotonicity of Perelman reduced volume on flows on compact Ricci flows (and more generally for Ricci flows of complete manifolds of bounded curvature). For instance, one can show (using compactness arguments in various weighted Sobolev spaces) that, as long as the manifold M is connected, a minimiser to (3) always exists, and is attained by an ${\mathcal L}$ -geodesic (defined as a curve $\gamma: [0,\tau_1] \to M$ for which the G quantity defined in (27) vanishes). [One can easily reduce to the connected case, since the reduced length is clearly infinite when $x_0$ and $x_1$ lie on distinct connected components.] Such geodesics turn out to have a well-defined “initial velocity” $v := \lim_{\tau \to 0} \sqrt{\tau} X(\tau)$ , as can be seen by working out the ODE for the quantity $\sqrt{\tau} X(\tau)$ (it is also convenient to reparameterise in terms of the variable $r := \sqrt{\tau}$ to remove any apparent singularity at $\tau=0$ ). This leads to an ${\mathcal L}$ -exponential map $\exp_{(0,x_0),\tau_1}: T_{x_0} M \to M$ for any fixed time $-\tau_1$ , which is smooth. The derivative of this map is controlled by ${\mathcal L}$ -Jacobi fields, which are close analogues of their elliptic counterparts, and which lead to the notion of a ${\mathcal L}$ -conjugate point $x_1$ to $(0,x_0)$ at the fixed time $-\tau_1$ . One can then define the injectivity domain and cut locus as before (again for a fixed time $-\tau_1$ ), and show as before that the former region has full measure. This lets one rigorously derive (6) almost everywhere (especially after noting that any segment of a minimising ${\mathcal L}$ -geodesic without conjugate points is again a minimising ${\mathcal L}$ -geodesic without conjugate points, thus establishing that the injectivity region is in some sense “star-shaped”), but again one needs to justify (6) in the sense of distributions in order to derive the monotonicity of Perelman reduced volume. This can again be done by use of barrier functions, perturbing the base point $(0,x_0)$ both spatially and also backwards in time by an epsilon. The details of this become rather technical; see for instance the paper of Ye or the notes of Kleiner-Lott, for details.

Thus far we have only discussed how reduced length and reduced volume behave on smooth Ricci flows of compact manifolds. Of course, to fully establish the global existence of Ricci flow with surgery, one also needs to build an analogous theory for Ricci flows with surgery. Here there turns out to be significant new technical difficulties, basically because one has to restrict attention to paths $\gamma$ which avoid all regions in which surgery is taking place. This creates some “holes” in the region of integration for the reduced volume, as in some cases the minimising path between two points in spacetime goes through a surgery region. Fortunately it turns out that (very roughly speaking) these holes only occur when the reduced length (or a somewhat technical modification thereof) is rather large, which means that the holes do not significantly impact lower bounds on this reduced volume, which is what is needed to establish $\kappa$ -noncollapsing. If time permits, I will discuss this issue further in later lectures, once we have described surgery in more detail.

In order to control ancient $\kappa$ -noncollapsing solutions, which are complete but not necessarily compact, one also needs to extend the above theory to complete non-compact manifolds. It turns out that this can be done as long as one has uniform bounds on curvature; a key task here is to establish that the reduced length $l_{(0,x_0)}(-\tau_1,x_1)$ behaves roughly like $d(x_0,x_1)^2/4\tau_1$ (which is basically what it is in the Euclidean case) as $x_1$ goes to infinity, which allows the integrand in the definition of reduced volume to have enough decay to justify all computations. The technical details here can be found in several places, including the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.

Remark 1. A theory analogous to Perelman’s theory above was worked out earlier by Li and Yau, but with the Ricci flow replaced by a static manifold with a lower bound on Ricci curvature, and with a time-dependent potential attached to the Laplacian. $\diamond$

by Terence Tao at May 09, 2008 03:40 PM

Terence Tao — 285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume

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

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);
the standard hyperbolic space $\sqrt{-K} \cdot H^d$ of constant sectional curvature K if $K < 0$ (Example 2 from Lecture 7); or
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

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May 09, 2008

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Terence Tao — 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume

Terence Tao — 285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume