What's new http://terrytao.wordpress.com Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Wed, 14 May 2008 15:42:39 +0000 http://wordpress.org/?v=MU en 285G, Lecture 11: κ-noncollapsing via Perelman reduced volume http://terrytao.wordpress.com/2008/05/14/285g-lecture-11-%ce%ba-noncollapsing-via-perelman-reduced-volume/ http://terrytao.wordpress.com/2008/05/14/285g-lecture-11-%ce%ba-noncollapsing-via-perelman-reduced-volume/#comments Wed, 14 May 2008 15:42:39 +0000 Terence Tao http://terrytao.wordpress.com/?p=378

Having established the monotonicity of the Perelman reduced volume in the previous lecture (after first heuristically justifying this monotonicity in Lecture 9), we now show how this can be used to establish \kappa-noncollapsing of Ricci flows, thus giving a second proof of Theorem 2 from Lecture 7. Of course, we already proved (a stronger version) of this theorem already in Lecture 8, using the Perelman entropy, but this second proof is also important, because the reduced volume is a more localised quantity (due to the weight e^{-l_{(0,x_0)}} in its definition and so one can in fact establish local versions of the non-collapsing theorem which turn out to be important when we study ancient \kappa-noncollapsing solutions later in Perelman’s proof, because such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy).

The route to \kappa-noncollapsing via reduced volume proceeds by the following scheme:

Non-collapsing at time t=0 (1)

\Downarrow

Large reduced volume at time t=0 (2)

\Downarrow

Large reduced volume at later times t (3)

\Downarrow

Non-collapsing at later times t (4)

The implication (2) \implies (3) is the monotonicity of Perelman reduced volume. In this lecture we discuss the other two implications (1) \implies (2), and (3) \implies (4)).

Our arguments here are based on Perelman’s first paper, Kleiner-Lott’s notes, and Morgan-Tian’s book, though the material in the Morgan-Tian book differs in some key respects from the other two texts. A closely related presentation of these topics also appears in the paper of Cao-Zhu.

– Definitions –

Let us first recall our definitions. Previously we defined Perelman reduced length and reduced volume for ancient flows t \mapsto (M,g(t)) for t \in (-\infty,0], centred at a point (0,x_0) on the final time slice t=0, but one can also define these quantities for flows on the time interval {}[0,T] and for points (t_0,x_0) \in [0,T] \times M as follows. We introduce the backward time variable \tau := t_0 - t. Given any path \gamma: [0,\tau_1] \to M, we define its length

\displaystyle L(\gamma) := \int_0^{\tau_1} \sqrt{\tau} (R + |\dot \gamma(\tau)|_g^2)\ d\tau(5)

and for any (t_1,x_1) with 0 \leq t_1 < t_0, with \tau_1 := t_0 - t_1, we define the reduced length

l_{(t_0,x_0)}(t_1,x_1) := \frac{1}{2\sqrt{\tau_1}} \inf_\gamma L(\gamma) (6)

where \gamma: [0,\tau_1] \to M ranges over all C^1 paths from x_0 to x_1 (which can also be viewed as trajectories in the spacetime manifold {}[0,T] \times M from (t_0,x_0) to (t_1,x_1). The reduced volume is then defined as

\displaystyle \tilde V_{(t_0,x_0)}(\tau_1) := \frac{1}{\tau_1^{d/2}} \int_M e^{-l_{(t_0,x_0)}(t_1,x_1)}\ d\mu_{t_1}(x_1). (7)

[Note: some authors normalise the reduced volume by using (4\pi \tau_1)^{d/2} instead of \tau_1^{d/2}, in order to give Euclidean space a reduced volume of 1, but this makes no essential difference to the analysis.]

The arguments of the previous lecture show that if t \mapsto (M,g(t)) is a Ricci flow, then the reduced volume is a non-increasing function of \tau_1 for fixed (t_0,x_0). In particular, the reduced volume at later times t_1 is bounded from below by the reduced volume at time 0 (which is the implication (2) \implies (3)).

– Heuristic analysis –

In the case of the trivial Euclidean flow, the reduced length is given by the formula

l_{(t_0,x_0)}(t_1,x_1) = \frac{|x_1-x_0|^2}{4\tau_1} = \frac{|x_1-x_0|^2}{4(t_1-t_0)} (8)

with the minimising geodesic given by the formula

\gamma(\tau) = x_0 + 2v \sqrt{\tau} with v := \frac{x_1-x_0}{2\sqrt{\tau_1}} (9)

Here, we briefly argue why we expect heuristically to have a similar relationship

l_{(t_0,x_0)}(t_1,x_1) \approx \frac{d_{g(t_1)}( x_0, x_1 )^2}{\tau_1} + O(1) (10)

for the reduced length on more general Ricci flows, under an assumption of bounded normalised curvature.

Specifically, suppose that we have a normalised curvature bound |\hbox{Riem}|_g = O(1/\tau_1). Then we have \dot g = - 2\hbox{Ric} = O( g / \tau_1 ), and so over the time scale \tau_1, we see that the metric only changes by a multiplicative constant. If we ignore such constants for now, we see that the distance function d_{g(t)}(x, y) does not change much over the time interval of interest.

Let \gamma be a minimising {\mathcal L}-geodesic from (t_0,x_0) to (t_1,x_1). This path has to traverse a distance roughly d_{g(t_1)}(x_0,x_1) in time \tau_1, and so its speed |\dot \gamma|_g should be at least d_{g(t_1)}(x_0,x_1) / \tau_1. Also, the scalar curvature R should be O(1/\tau_1) by the bounded normalised curvature assumption. Putting all this into (5) and (6) we heuristically obtain (10).

From (10), we expect the expression e^{-l_{(t_0,x_0)}(t_1,x_1)} to be comparable to 1 when x_1 is inside the ball B_{g(t_1)}(x_0, O( \sqrt{t_1} )), and to be exponentially small outside of this ball. Using (7), we thus obtain a heuristic approximation for the Perelman reduced volume:

\tilde V_{(t_0,x_0)}(\tau_1) \approx \hbox{Vol}_{g(t_1)}(x_0, \sqrt{\tau_1} ) / \tau_1^{d/2}. (11)

Thus the Perelman reduced volume \tilde V_{(t_0,x_0)}(\tau_1) is heuristically equivalent to the Bishop-Gromov reduced volume at (x_1,t_1) at scale \tau_1. Since the latter measures non-collapsing, we heuristically obtain the implications (1) \implies (2) and (3) \implies (4).

– From non-collapsing to lower bounds on reduced volume –

Now we discuss implications of the form (1) \implies (2) in more detail. Specifically, we show

Proposition 1. Let t \mapsto (M,g(t)) be a d-dimensional Ricci flow on a complete manifold M for t \in [0,T] such that we have the normalised initial conditions |\hbox{Riem}(0,x)|_g \leq 1 and \hbox{Vol}_{g(0)}(B_{g(0)}(x,1)) \geq \omega at time t=0 for some \omega > 0 and all x (so in particular, the geometry is non-collapsed at scale 1 at all points at time zero). Then we have \tilde V_{(t_0,x_0)}(t_0) \geq c for some c = c(d,\omega,T) > 0 and all (t_0,x_0) \in (0,T) \times M.

The main task in proving implications of the form (1) \implies (2) is to show the existence of some large ball at time zero on which l = l_{(t_0,x_0)} is bounded from above.

Turning to the specific proposition above, we first observe that we can reduce to the large time case t_0 \geq 1. Indeed, if 0 < t_0 < 1, then we can rescale the Ricci flow until t_0 = 1 (this increases T, but we can simply truncate T to compensate for this). This rescaling reduces the size of the initial Riemann curvature, and the volume of balls of unit radius are still bounded from below thanks to the Bishop-Gromov inequality.

The next observation we need is that the control on the geometry at time zero persists for a short amount of additional time:

Lemma 1. (Local persistence of controlled geometry) Let the hypotheses be as in Proposition 1. Then there exists an absolute constant c > 0 (depending only on d) such that |\hbox{Riem}(t,x)|_g \leq 2 for all times 0 \leq t \leq c and x \in M. Also we have \hbox{Vol}_{g(t)}(B_{g(t)}(x,1)) \ge \omega' for all 0 \leq t \leq c and x \in M, and some \omega' > 0 depending only on \omega, d.

Proof. We recall the nonlinear heat equation

\partial_t \hbox{Riem} = \Delta \hbox{Riem} + {\mathcal O}( g^{-1} \hbox{Riem}^2 ) (12)

for the Riemann curvature tensor \Box under Ricci flow (see equation (31) of Lecture 1). The bound on Riemann curvature can then obtained by an application of Hamilton’s maximum principle (Proposition 1 from Lecture 3); we leave this as an exercise to the reader. As in the heuristic discussion, the bounds on the Riemann curvature (and hence the Ricci curvature) show that the metric g and the distance function d_{g(t)}(x,y) only change by at most a multiplicative constant; this also implies that the volume measure only changes by a multiplicative constant as well. From this we see that the lower bound on the volume of unit balls at time zero implies a lower bound on the volume of balls of radius O(1) at times 0 \leq t \leq c; one can then get back to balls of radius 1 by invoking the Bishop-Gromov inequality. \Box

The next task is to find a point y \in M such that the reduced length from (t_0,x_0) to (0,y) is small, since this should force y (and the points close to y) to give a large contribution to the reduced volume. In the Euclidean case, one would just take y = x_0 (see (8)), but this does not necessarily work for general Ricci flows: note from (5), (6) that the reduced length from (t_0,x_0) to (t_1,x_0) could in principle be as large as

\displaystyle \frac{1}{2\sqrt{t_0-t_1}} \int_0^{t_0-t_1} \sqrt{\tau} R(t_0-\tau,x_0)\ d\tau, (13)

which could be quite large if the scalar curvature becomes large and positive (which is certainly within the realm of possibility, especially if one is approaching a singularity).

Fortunately, we can use the parabolic properties of the reduced length l = l_{(t_0,x_0)}, combined with the maximum principle, to locate a good point y with the required properties. From the analysis of the previous lecture, and some rescaling and time translation, we obtain the identities and inequalities

\nabla l = X (14)

\partial_\tau l = \frac{1}{2} R - \frac{1}{2} |X|_g^2 - \frac{1}{2\tau} l (15)

\Delta l \leq \frac{d}{2\tau} + \frac{1}{2} |X|_g^2 - \frac{1}{2} R - \frac{1}{2\tau} l (16)

(cf. equations (29), (33), (47) from the previous lecture), where X = \gamma'(\tau) is the final velocity vector of the minimising {\mathcal L}-geodesic from (t_0,x_0) to (t_1,x_1). [We only derived (14)-(16) rigorously inside the domain of injectivity, but as discussed in the previous lecture, one can establish the above inequalities in the sense of distributions on the whole manifold M.] From (15), (16) we obtain in particular that l is a supersolution of a heat equation:

\partial_t l \geq \Delta l + \frac{l-(d/2)}{\tau}. (17)

[Note that (17) holds with equality in the Euclidean case (8).] From the maximum principle (Corollary 1 from Lecture 3), we see that if we have the uniform lower bound l \geq d/2 at some time 0 \leq t < t_0, then this bound will persist for all times between t and t_0. On the other hand, by using the upper bound (12) for l(t_1,x_0) we see that the bound l \geq d/2 breaks down for times t sufficiently close to t_0. We therefore conclude that \inf_{x \in M} l(t,x) < d/2 for all 0 \leq t < t_0. In particular we can find a point y such that

l(c,y) < d/2, (18)

where c is the small constant in Lemma 1. Given the bounded geometry control in Lemma 1 (and in particular the fact that g(t) is comparable to g(0) for 0 \leq t \leq c), it is thus not hard to see (by concatenating the minimising path from (0,x_0) to (c,y) with a geodesic segment (in the g(0) metric) from (c,y) to (0,y')) that

l(0, y') \leq C \hbox{ for } y' \in B_{g(0)}(y, c') (19)

for some C, c' > 0 depending only on d, where. The hypotheses on the geometry of g(0), combined with the Bishop-Gromov inequality, give a uniform lower bound for the volume of B_{g(0)}(y,c'), and Proposition 1 now follows directly from the definition (7) of reduced volume.

– From lower bounds on reduced volume to non-collapsing –

Now we consider the reverse type of implication (3) \implies (4) from those just discussed. Here, the task is reversed; rather than establishing upper bounds on l on a ball of radius comparable to one, the main challenge is now to establish lower bounds (of the form l \geq -O(1)) on l on such a ball, as well as some growth bounds on l away from this ball.

We begin by formally stating the result of the form (3) \implies (4) that we shall establish.

Proposition 2. Let t \mapsto (M,g(t)) be a d-dimensional Ricci flow on a complete manifold M for t \in [0,T], and let 0 \leq t_0-r_0^2 \leq t_0 \leq T and x_0 \in M be such that |\hbox{Riem}(t,x)|_g \leq r_0^{-2} for x \in B_{g(t_0)}(x_0,r_0) and t \in [t_0-r_0^2,t_0], and such that \tilde V_{(t_0,x_0)}(\tau) \geq \delta for some \delta > 0 and all 0 < \tau < r_0^2. Then one has \hbox{Vol}_{g(t_0)}( B_{g(t_0)}(x_0,r_0) ) \geq c for some c depending only on d and \delta.

Exercise 1. Use Proposition 1, Proposition 2, and the monotonicity of Perelman reduced volume to deduce Theorem 2 from Lecture 7. \diamond

We now prove Proposition 2. We first observe by time translation (and by removing the portion of the Ricci flow below t_0-r_0^2 that we may normalise t_0-r_0^2=0, and then by scaling we may normalise t_0 = 1. Thus we now have a Ricci flow on [0,1] with |\hbox{Riem}(t,x)|_g \leq 1 on {}[0,1] \times B_{g(1)}(x_0,1) and

\displaystyle \tilde V_{(1,x_0)}(\tau) = \int_M e^{-l(\tau,x)}\ d\mu_{g(\tau)}(x) \geq \delta (20)

for all 0 < \tau \leq 1, where l = l_{(1,x_0)} is the reduced length function. Our task is to show that \hbox{Vol}_{g(1)}( B_{g(1)}(x_0,1) ) is bounded away from zero.

We first observe (as in Lemma 1) that the metrics g(t) for 0 \leq t \leq 1 are all comparable to each other up to multiplicative constants on B_{g(1)}(x_0,1), and so the balls in these metrics also differ only up to multiplicative constants.

Next, we would like to localise the reduced volume (20) to the ball B_{g(1)}(x_0,1) (since this is the only place where we really control the geometry). To do this it is convenient to work in the parabolic counterpart of normal coordinates around (1,x_0) and exploit the pointwise version of the Perelman reduced volume monotonicity. To motivate this, recall from the pointwise inequality

{\mathcal L}_{\partial r} d\mu \leq \frac{d-1}{r}\ d\mu (21)

that we had the Bishop-Gromov inequality

\displaystyle \partial_r r^{-(d-1)} \int_{S(x_0,r)}\ dS \leq 0 (21′)

where S(x_0,r) is the sphere of radius r centred at x_0 with area element dS. Indeed, we can rewrite the left-hand side of (21′) as

\displaystyle \partial_r r^{-(d-1)} \int_{S^{d-1}} J_r( \omega )\ d\omega (22)

where S^{d-1} is the standard sphere with the standard area element d\omega, and J_r is the Jacobian of the exponential map \omega \mapsto \exp_{x_0}(r \omega); in the Euclidean case, J_r(\omega) = r^{d-1}. [Actually, once the radius r exceeds the injectivity radius, one has to restrict to the portion of S^{d-1} that has not yet encountered the cut locus, but let us ignore this technical issue for now.] The inequality (21) (when combined with the Gauss lemma) is equivalent to the pointwise inequality

\partial_r r^{-(d-1)} J_r(\omega) \leq 0 (23)

which certainly implies (22), but also implies the stronger fact that the Bishop-Gromov inequality can be localised to arbitrary sectors in the sense that r^{-(d-1)} \int_{\Omega} J_r(\omega)\ d\omega (which can be viewed as the Bishop-Gromov reduced volume of the sector \{ \exp_{x_0}(r\omega): \omega \in \Omega \}) is non-increasing in r.

Now we develop parabolic analogues of the above observations. Recall from the previous lecture that we have an {\mathcal L}-exponential map {\mathcal L}\exp_{(1,x_0),\tau_1}: T_{x_0} M \to M for 0 \leq \tau_1 \leq 1 that sends a tangent vector v to \gamma(\tau_1), where \gamma: [0,\tau] \to M is the unique {\mathcal L}-geodesic starting at x_0 with initial condition v = \lim_{\tau \to 0} \sqrt{\tau} X(\tau) = \lim_{\tau \to 0} \sqrt{\tau} \gamma'(\tau). In the Euclidean case, this map is given by the formula

{\mathcal L}\exp_{(1,x_0),\tau_1}(v)= x_0 + 2 (x_1-x_0) \sqrt{\tau_1} (24)

as can be seen from (9). We can then rewrite the reduced volume \tilde V_{(1,x_0)}(\tau) in terms of “normal coordinates” as

\tilde V_{(1,x_0)}(\tau) = \tau^{-d/2} \int_{{\Bbb R}^d} e^{-l({\mathcal L} \exp_{(1,x_0),\tau}(v))} J_{\tau}(v)\ dv (25)

where J_\tau is the Jacobian of the map v \mapsto {\mathcal L}\exp_{(1,x_0),\tau_1}(v). (Again, one has to restrict {\Bbb R}^d to the portion of the tangent manifold lies inside the injectivity domain, but this domain turns out to be non-increasing in \tau (for much the same reason that the region inside the cut locus of a point in a Riemannian manifold is star-shaped) and so this effect works in our favour as far as monotonicity is concerned.)

In the previous lecture we saw that the monotonicity of Perelman reduced volume followed from the pointwise inequality

\displaystyle \partial_{\tau} l - \Delta l + |\nabla l|_{g}^2 - R + \frac{d}{2\tau} \geq 0 (26)

which of course also follows from (14)-(16).

Exercise 2. Use (14), (26), and the identity

\partial_\tau \exp_{(1,x_0),\tau}(v) = X (27)

(which basically follows from the fact that any segment of a minimising {\mathcal L}-geodesic is again a {\mathcal L}-geodesic) to derive the pointwise inequality

\displaystyle \partial_\tau \tau^{-d/2} e^{-l({\mathcal L} \exp_{(1,x_0),\tau}(v))} J_{\tau}(v) \leq 0. \diamond (38)

Exercise 2 reproves the monotonicity of Perelman reduced volume (25), but also proves a stronger local version of this monotonicity in which the region of integration {\mathbb R}^d is replaced by an arbitrary region \Omega (intersected with the injectivity region, as mentioned earlier).

In the Euclidean case, a computation using (8) and (24) shows that l({\mathcal L} \exp_{(1,x_0),\tau}(v)) = |v|^2 and J_\tau(v) = 2^n \tau^{d/2}. Also, one can use some basic analysis arguments to show that in the limit \tau \to 0, the expressions in (25) converge pointwise to their Euclidean counterparts. As a consequence we obtain the pointwise domination

\displaystyle \tau^{-d/2} e^{-l({\mathcal L} \exp_{(1,x_0),\tau}(v))} J_{\tau}(v) \leq 2^{n/2} e^{-|v|^2} (39)

for any v and any 0 < \tau < 1. As a consequence, the far part of (25) (corresponding to “fast” geodesics) is negligible: we have

\displaystyle \tau^{-d/2} \int_{|v| > C} e^{-l({\mathcal L} \exp_{(1,x_0),\tau}(v))} J_{\tau}(v)\ dv \leq \delta/2 (40)

for some C depending only on d and \delta. From this and the hypothesis (19) we thus obtain lower bounds on local Perelman reduced volume, or more precisely that

\displaystyle \tau^{-d/2} \int_{|v| \leq C} e^{-l({\mathcal L} \exp_{(1,x_0),\tau}(v))} J_{\tau}(v)\ dv \geq \delta/2 (41)

for all 0 < \tau \leq 1.

Now, we have bounded curvature on the cylinder {}[0,1] \times B_{g(1)}(x_0,1). Using the heat equation (12) and standard parabolic regularity estimates, we thus conclude that any first derivatives of the curvature are also bounded on the cylinder {}[1/2,1] \times B_{(g(1)}(x_0,1/2). (In fact, all higher derivatives are controlled as well; see this paper of Shi for full details.) In particular we have \nabla R = O(1) in this cylinder. Thus the equation G=0 for an {\mathcal L}-geodesic (where G was defined in equation (27) of the previous lecture) becomes

\nabla_\tau X + \frac{1}{2\tau} X = O(1) + O( |X| ) (42)

or equivalently that

\nabla_\tau (\sqrt{\tau} X) = O(\sqrt{\tau}) + O( \sqrt{\tau} |X| ) (43)

as long as the geodesic stays inside this smaller cylinder. From this and Gronwall’s inequality one easily verifies that for sufficiently small 0 < \tau < 1/2 (depending on C, d), the exponential map {\mathcal L}\exp_{(1,x_0),\tau}(v) does not exit the cylinder {}[1/2,1] \times B_{g(1)}(x_0,1/2) for |v |\leq C. On the other hand, at time \tau, we see from (5), (6) and the bounds on curvature in this cylinder that the reduced length l of the associated {\mathcal L}-geodesic is bounded below by some constant depending on \tau, C, d. We thus see (from the change of variables formula) that the left-hand side of (41) is bounded above by O_{\tau,C,d}( \hbox{Vol}_{g(1-\tau)}( B_{g(1)}(x_0,1/2) ) ). Choosing \tau to be a small number depending on C, d, we thus conclude from (41) that the volume of B_{g(1)}(x_0,1) with respect to g(1-\tau) (and hence g(1), by comparability of metrics) is bounded from below by some constant depending on C and d, and thus ultimately on \delta and d, giving Proposition 2 as desired.

– Extensions –

The pointwise nature of the monotonicity of Perelman reduced volume allows one to derive local versions of the non-collapsing result, in which one only needs a portion of the geometry to be non-collapsed at the initial time. A typical version of such a local noncollapsing result reads as follows.

Theorem 1 (Perelman’s non-collapsing theorem, second version) Let t \mapsto (M,g(t)) be a d-dimensional Ricci flow on the time interval {}[0,r_0^2], and suppose that one has the bounded normalised curvature condition |\hbox{Riem}|_g \leq r_0^{-2} on a cylinder {}[0,r_0^2] \times B_{g(0)}(x_0,r_0) for some x_0 \in M. Suppose also that we have the volume lower bound \hbox{Vol}_{g(0)}(B_{g(0)}(x_0,r_0)) \geq c r_0^d for some c>0. Then for any A > 0, the Ricci flow is \kappa-noncollapsed at (r_0^2,x) for any x \in B_{g(r_0^2)}(x_0,Ar_0) and at any scale 0 < r < r_0, for some \kappa depending only on d, c, A.

The novelty here is that the geometry is controlled in a cylinder, rather than on the initial time slice, but one gets to conclude \kappa-noncollapsing at points some distance away from the cylinder. In view of Lemma 1, we see that this result is more or less a strengthening of the previous \kappa-noncollapsing theorem.

This theorem is used in the original argument of Perelman (and then in the later treatments by Kleiner-Lott and Cao-Zhu) but appears to have been avoided in the treatment of Morgan-Tian, who instead rely on a variant of the original non-collapsing theorem in which surgery is permitted. At present I have not fully understood where this discrepancy comes from, but I tentatively am inclined to agree with Morgan-Tian that this local non-collapsing theorem is not essential to the argument. Nevertheless I will sketch how the above theorem is proven below, since there are one or two interesting technical tricks that get used in the argument.

The proof of Theorem 1 is, unsurprisingly, a modification of the previous arguments . The implications (2) \implies (3) and (3) \implies (4) are basically unchanged, but one needs to replace Proposition 1 by the following variant.

Proposition 3. Let the hypotheses be as in Theorem 1. Then for any x \in B_{g(r_0^2)}(x_0,Ar_0) one has \tilde V_{(r_0^2,x)}(r_0^2) \geq c' for some c' > 0 depending on A, c, d.

We sketch the proof of Proposition 3. It is convenient to rescale so that r_0=1. In view of the non-collapsed nature of the geometry in B_{g(0)}(x_0,1), it suffices to establish a lower bound of the form l_{(1,x)}(0,z) \geq -C for all z \in B_{g(0)}(x_0,1/2) for some C > 0 depending on A,c,d. Actually, because of the bounded geometry in the cylinder, it suffices to show that l_{(1,x)}(1/2,y) \geq -C' for just one point z \in B_{g(1/2)}(x_0,1/10) for some C' > 0 depending on A,c,d, since one can join (1/2,y) by a geodesic to (1,z) much as in the proof of Proposition 1.

The task is now analogous to that of finding a point y that obeyed the relation (18), so we expect the heat equation (17) to again play a role. We do not need the sharp bound of n/2 which occurs in (18); on the other hand, y is now constrained to lie in a ball, which defeats a direct application of the maximum principle. To fix this one has to multiply the reduced length l by a penalising weight to force the minimum to lie in the desired ball at time 1/2, and then rapidly relax this weight as one moves from time 1/2 to time 1 so that it incorporates the point x at time 1. It turns out the maximum principle can then be applied with a suitable choice of weights, as long as one knows that the distance function r(t,y) = d_{g(t)}(x_0,y) is a supersolution to a heat equation, and more precisely that \partial_t r - \Delta r \geq -C when r is bounded away from the origin. But this can be established by the first and second variation formulae for the distance function, and in particular using the non-negativity of the second variation for minimising geodesics. Details can be found in Section 8 of Perelman’s paper, Sections 26-27 of Kleiner-Lott, or Section 3.4 of Cao-Zhu.

Remark 1. One can also interpret the above analysis in terms of heat kernels, and using (26) instead of (17). The former inequality is equivalent to the assertion that the function v := (4\pi \tau)^{-d/2} e^{-l} is a subsolution of the adjoint heat equation: \partial_t v + \Delta v - Rv \leq 0. As t \to 1, v approaches a Dirac mass at x (indeed, v asymptotically resembles the Euclidean backwards heat kernel from (1,x_0)) and the task is to obtain upper bounds on v at some point on a ball B_{g(1/2)}(x_0,1/10) at time 1/2. This is basically equivalent to establishing lower bounds of Gaussian type for the fundamental solution of the adjoint heat equation at some point in B_{g(1/2)}(x_0,1/10). Similar analysis in the case of a static manifold with potential (and a lower bound on Ricci curvature) was carried out somewhat earlier by Li and Yau. \diamond

As mentioned previously, in order to apply the non-collapsing result beyond the first surgery time, it is necessary to develop analogues of the above theory for Ricci flows with surgery. This turns out to be remarkably technical, but the main ideas at least are fairly clear. Firstly, one has to delete all {\mathcal L}-geodesics which pass through surgery regions when defining the Perelman reduced volume; such curves are called “inadmissible”. Note that if (1,x_0) is in a surgery region to begin with, then every curve is inadmissible but in this case the geometry can be controlled directly from the surgery theory. As it turns out, one can similarly deal with the case when (1,x_0) has extremely high curvature because one can control the geometry of such regions. So we can easily eliminate these bad cases.

Because of the pointwise nature of the monotonicity formula for reduced volume, this restriction of admissibility does not affect the “(2) \implies (3)” stage of the argument. The “(3) \implies (4)” step is also largely unaffected, since removing inadmissible components of the reduced volume only serves to strengthen the hypothesis (3). But significant new technical difficulties arise in the “(1) \implies (2)” portion of the argument, when one has to argue that not too much of the reduced volume has been deleted by all the various surgeries that take place between time t=0 and time t=1. In particular, we still need to find a point y obeying (18) (or something very much like (18)) which is admissible. To do this, the basic idea is to establish that inadmissible curves have large reduced length (and so removing them will not impact the search for a solution to (18)). For technical reasons it is better to restrict attention to barely admissible curves - curves which just touch the border of the surgery region, but do not actually enter it. In this case it is possible to use the geometric control of the surgery regions to give some non-trivial lower bounds on the reduced length of such curves, although there are still significant technical issues to resolve beyond this. I hope to return to this point later in the course, when we have defined surgery properly.

– Epilogue: a connection between Perelman entropy and Perelman reduced volume –

We have shown two routes towards establishing \kappa-non-collapsing of Ricci flows, one using the (parameterised) Perelman entropies

\displaystyle \mu(g(t),\tau) := \inf \{ \int_M (\tau(|\nabla f|^2 + R) + f - d) (4\pi\tau)^{-d/2} e^{-f}\ d\mu:

\displaystyle \int_M (4\pi\tau)^{-d/2} e^{-f}\ d\mu = 1 \} (44)

and one using the reduced volumes \tilde V_{(0,x_0)} mentioned above. Actually, the two quantities are related to each other (this is hinted at in Section 9 of Perelman’s paper); very roughly speaking, the potential function f in the theory of Perelman entropy plays the same role that reduced length l does in the theory of Perelman volume. Indeed, using (44) and shifting f by a constant if necessary, we have the log-Sobolev inequality

\displaystyle \int_M (\tau(|\nabla f|^2 + R) + f - d) (4\pi\tau)^{-d/2} e^{-f}\ d\mu

\displaystyle \geq [\mu(g(t),\tau) - \log \int_M (4\pi\tau)^{-d/2} e^{-f}\ d\mu] \int_M (4\pi\tau)^{-d/2} e^{-f}\ d\mu. (45)

An integration by parts reveals that we can replace the |\nabla f|^2 on the left -hand side by \Delta f, and hence one can also replace this quantity by 2\Delta f - |\nabla f|^2.

We now apply this inequality with \tau := t_0-t and f = l_{(t_0,x_0)} for some spacetime point (t_0,x_0) in the Ricci flow. Using (14), (16) we see that

2 \Delta f - |\nabla f|^2 \leq \frac{d}{2\tau} - R - \frac{1}{\tau} f (46)

and thus the left-hand side of (45) is non-positive. Using (7) we thus conclude a simple relationship between entropy and reduced volume:

\mu(g(t_0-\tau),\tau) \leq \log \frac{\tilde V_{(t_0,x_0)}(\tau)}{(4\pi)^{d/2}}. (47)

[As usual, we have equality in physical space; this inequality also reinforces the suggestion that one normalise the reduced volume by an additional factor of 1/(4\pi)^{d/2}.]

Thus the Perelman entropy can be viewed as a global analogue of the Perelman reduced volume, in which we allow the base point x_0 to vary (thus it measures the global non-collapsing nature of the manifold, as opposed to the local nature; we already saw this in Lecture 8; compare in particular equation (62) from Lecture 8 with the heuristic (11) using (47).)

There are other connections between entropy and reduced volume; compare for instance the flow equation for the potential f (equation (46) from Lecture 8) with equation (26) here. The adjoint heat equation \partial_t u + \Delta u - Ru = 0 also makes essentially the same appearance in both theories. See Section 9 of Perelman’s paper for further discussion.

Remark 2. As remarked above, the flow equation for f can be viewed as a pointwise versions of the entropy monotonicity formula, which in principle leads to localised monotonicity formulae for the Perelman entropy; some analysis in this direction appears in Section 9 of Perelman’s paper. But I do not know if these localised entropy formulae can substitute to give a different proof of Theorem 1. \diamond

]]> http://terrytao.wordpress.com/2008/05/14/285g-lecture-11-%ce%ba-noncollapsing-via-perelman-reduced-volume/feed/ Terry A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential http://terrytao.wordpress.com/2008/05/13/a-global-compact-attractor-for-high-dimensional-defocusing-non-linear-schrodinger-equations-with-potential/ http://terrytao.wordpress.com/2008/05/13/a-global-compact-attractor-for-high-dimensional-defocusing-non-linear-schrodinger-equations-with-potential/#comments Tue, 13 May 2008 19:04:34 +0000 Terence Tao http://terrytao.wordpress.com/?p=377

I’ve just uploaded to the arXiv my paper “A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential“, submitted to Dynamics of PDE. This paper continues some earlier work of myself in an attempt to understand the soliton resolution conjecture for various nonlinear dispersive equations, and in particular, nonlinear Schrödinger equations (NLS). This conjecture (which I also discussed in my third Simons lecture) asserts, roughly speaking, that any reasonable (e.g. bounded energy) solution to such equations eventually resolves into a superposition of a radiation component (which behaves like a solution to the linear Schrödinger equation) plus a finite number of “nonlinear bound states” or “solitons”. This conjecture is known in many perturbative cases (when the solution is close to a special solution, such as the vacuum state or a ground state) as well as in defocusing cases (in which no non-trivial bound states or solitons exist), but is still almost completely open in non-perturbative situations (in which the solution is large and not close to a special solution) which contain at least one bound state. In my earlier papers, I was able to show that for certain NLS models in sufficiently high dimension, one could at least say that such solutions resolved into a radiation term plus a finite number of “weakly bound” states whose evolution was essentially almost periodic (or almost periodic modulo translation symmetries). These bound states also enjoyed various additional decay and regularity properties. As a consequence of this, in five and higher dimensions (and for reasonable nonlinearities), and assuming spherical symmetry, I showed that there was a (local) compact attractor K_E for the flow: any solution with energy bounded by some given level E would eventually decouple into a radiation term, plus a state which converged to this compact attractor K_E. In that result, I did not rule out the possibility that this attractor depended on the energy E. Indeed, it is conceivable for many models that there exist nonlinear bound states of arbitrarily high energy, which would mean that K_E must increase in size as E increases to accommodate these states. (I discuss these results in a recent talk of mine.)

In my new paper, following a suggestion of Michael Weinstein, I consider the NLS equation

i u_t + \Delta u = |u|^{p-1} u + Vu

where u: {\Bbb R} \times {\Bbb R}^d \to {\Bbb C} is the solution, and V \in C^\infty_0({\Bbb R}^d) is a smooth compactly supported real potential. We make the standard assumption 1 + \frac{4}{d} < p < 1 + \frac{4}{d-2} (which is asserting that the nonlinearity is mass-supercritical and energy-subcritical). In the absence of this potential (i.e. when V=0), this is the defocusing nonlinear Schrödinger equation, which is known to have no bound states, and in fact it is known in this case that all finite energy solutions eventually scatter into a radiation state (which asymptotically resembles a solution to the linear Schrödinger equation). However, once one adds a potential (particularly one which is large and negative), both linear bound states (solutions to the linear eigenstate equation (-\Delta + V) Q = -E Q) and nonlinear bound states (solutions to the nonlinear eigenstate equation (-\Delta+V)Q = -EQ - |Q|^{p-1} Q) can appear. Thus in this case the soliton resolution conjecture predicts that solutions should resolve into a scattering state (that behaves as if the potential was not present), plus a finite number of (nonlinear) bound states. There is a fair amount of work towards this conjecture for this model in perturbative cases (when the energy is small), but the case of large energy solutions is still open.

In my new paper, I consider the large energy case, assuming spherical symmetry. For technical reasons, I also need to assume very high dimension d \geq 11. The main result is the existence of a global compact attractor K: every finite energy solution, no matter how large, eventually resolves into a scattering state and a state which converges to K. In particular, since K is bounded, all but a bounded amount of energy will be radiated off to infinity. Another corollary of this result is that the space of all nonlinear bound states for this model is compact. Intuitively, the point is that when the solution gets very large, the defocusing nonlinearity dominates any attractive aspects of the potential V, and so the solution will disperse in this case; thus one expects the only bound states to be bounded. The spherical symmetry assumption also restricts the bound states to lie near the origin, thus yielding the compactness. (It is also conceivable that the localised nature of V also restricts bound states to lie near the origin, even without the help of spherical symmetry, but I was not able to establish this rigorously.)

In view of my previous results concerning local compact attractors, the main difficulty is to show that spherically symmetric almost periodic solutions - solutions which range inside a compact subset of the energy space - enjoy a universal upper bound on their energy and mass. (This can be viewed as a “quasi-Liouville theorem”, in analogy with other recent Liouville theorems in the literature which classify various types of almost periodic solutions.)

This is accomplished in two stages. Firstly, by extensive use of the Duhamel formula and the dispersive properties of the free Schrödinger propagator (as in my previous papers), one shows that spherically symmetric almost periodic solutions exhibit quite strong decay away from the origin (more than is predicted just from the finite energy hypothesis); indeed, they decay like the Newton potential |x|^{2-d} (which makes sense, if one looks at the bound state equation). In high dimension, this gives additional moment bounds on the solution. For instance, in 11 and higher dimensions, it implies that not only do almost periodic solutions have finite mass (which means that \int_{{\Bbb R}^d} |u(t,x)|^2\ dx is finite) but that the sixth moment \int_{{\Bbb R}^d} |u(t,x)|^2 |x|^6\ dx is also finite.

These moment conditions allow one to use some exotic virial identities. The basic virial identity for NLS is given by the formula

\partial_t \int_{{\Bbb R}^d} \nabla a \cdot \hbox{Im}( \overline{u} \nabla u )\ dx

= 2 \int_{{\Bbb R}^d} \hbox{Hess}(a)( \nabla u, \overline{\nabla u} )\ dx

+ \frac{p-1}{p+1} \int_{{\Bbb R}^d} |u|^{p+1} \Delta a\ dx

- \frac{1}{2} \int_{{\Bbb R}^d} |u|^2 \Delta \Delta a\ dx

- \int_{{\Bbb R}^d} (\nabla a \cdot \nabla V) |u|^2\ dx

where a: {\Bbb R}^d \to {\Bbb R} is a weight function which has to obey some reasonable regularity and growth hypotheses but is otherwise arbitrary. The more moment conditions one has on u, the more rapid one can take the growth of a to be.

Different choices of the weight a yield different interesting consequences. For instance, a(x):=1 gives the momentum conservation law, while a(x) := |x| gives the Morawetz inequalities. The choice a(x):=|x|^2 gives the virial identity of Glassey, which I use to establish a universal bound on the energy. It turns out that the choice a(x) := |x|^4 gives an identity that can give a universal bound on the mass (coming from the \Delta \Delta a term in the identity), which yields the main theorem; the dimension hypothesis d \geq 11 is needed to get enough decay on the almost periodic solution in order to justify the formal application of the virial identity with this quartic weight. (By working a bit harder I was able to weaken this hypothesis to d \geq 7, but the correct hypothesis should be d \geq 5, in analogy with the classical theory of resonances for the linear Schrödinger operator with potential.)

One technical feature that comes up when dealing with superquadratic weights such as |x|^4 is that the mass term that involves \Delta \Delta a is negative, which looks unfavourable. Fortunately, it turns out that one can use Hardy’s inequality and the term coming from the Hessian \hbox{Hess}(a) to convert this negative term into a positive one.

There is an amusing consequence of these results; once one has a global compact attractor for a PDE, it becomes possible in principle to establish soliton resolution for this PDE by a finite amount of rigorous numerics on that attractor (or on some larger compact set containing that attractor), combined with some quantitative nonlinear stability results on all the soliton states. However such a program would be extremely complicated to execute in practice.

]]> http://terrytao.wordpress.com/2008/05/13/a-global-compact-attractor-for-high-dimensional-defocusing-non-linear-schrodinger-equations-with-potential/feed/ Terry 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume http://terrytao.wordpress.com/2008/05/09/285g-lecture-10-variation-of-l-geodesics-and-monotonicity-of-perelman-reduced-volume/ http://terrytao.wordpress.com/2008/05/09/285g-lecture-10-variation-of-l-geodesics-and-monotonicity-of-perelman-reduced-volume/#comments Fri, 09 May 2008 15:40:23 +0000 Terence Tao http://terrytao.wordpress.com/?p=376

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-increasing 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-increasing 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-increasing 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