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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.

It is well known that the heat equation

$\dot f = \Delta f$ (1)

on a compact Riemannian manifold (M,g) (with metric g static, i.e. independent of time), where $f: [0,T] \times M \to {\Bbb R}$ is a scalar field, can be interpreted as the gradient flow for the Dirichlet energy functional

$\displaystyle E(f) := \frac{1}{2} \int_M |\nabla f|_g^2\ d\mu$ (2)

using the inner product $\langle f_1, f_2 \rangle_\mu := \int_M f_1 f_2\ d\mu$ associated to the volume measure $d\mu$. Indeed, if we evolve f in time at some arbitrary rate $\dot f$, a simple application of integration by parts (equation (29) from Lecture 1) gives

$\displaystyle \frac{d}{dt} E(f) = - \int_M (\Delta f) \dot f\ d\mu = \langle -\Delta f, \dot f \rangle_\mu$ (3)

from which we see that (1) is indeed the gradient flow for (3) with respect to the inner product. In particular, if f solves the heat equation (1), we see that the Dirichlet energy is decreasing in time:

$\displaystyle \frac{d}{dt} E(f) = - \int_M |\Delta f|^2\ d\mu$. (4)

Thus we see that by representing the PDE (1) as a gradient flow, we automatically gain a controlled quantity of the evolution, namely the energy functional that is generating the gradient flow. This representation also strongly suggests (though does not quite prove) that solutions of (1) should eventually converge to stationary points of the Dirichlet energy (2), which by (3) are just the harmonic functions (i.e. the functions f with $\Delta f = 0$).

As one very quick application of the gradient flow interpretation, we can assert that the only periodic (or “breather”) solutions to the heat equation (1) are the harmonic functions (which, in fact, must be constant if M is compact, thanks to the maximum principle). Indeed, if a solution f was periodic, then the monotone functional E must be constant, which by (4) implies that f is harmonic as claimed.

It would therefore be desirable to represent Ricci flow as a gradient flow also, in order to gain a new controlled quantity, and also to gain some hints as to what the asymptotic behaviour of Ricci flows should be. It turns out that one cannot quite do this directly (there is an obstruction caused by gradient steady solitons, of which we shall say more later); but Perelman nevertheless observed that one can interpret Ricci flow as gradient flow if one first quotients out the diffeomorphism invariance of the flow. In fact, there are infinitely many such gradient flow interpretations available. This fact already allows one to rule out “breather” solutions to Ricci flow, and also reveals some information about how Poincaré’s inequality deforms under this flow.

The energy functionals associated to the above interpretations are subcritical (in fact, they are much like $R_{\min}$) but they are not coercive; Poincaré’s inequality holds both in collapsed and non-collapsed geometries, and so these functionals are not excluding the former. However, Perelman discovered a perturbation of these functionals associated to a deeper inequality, the log-Sobolev inequality (first introduced by Gross in Euclidean space). This inequality is sensitive to volume collapsing at a given scale. Furthermore, by optimising over the scale parameter, the controlled quantity (now known as the Perelman entropy) becomes scale-invariant and prevents collapsing at any scale – precisely what is needed to carry out the first phase of the strategy outlined in the previous lecture to establish global existence of Ricci flow with surgery.

The material here is loosely based on Perelman’s paper, Kleiner-Lott’s notes, and Müller’s book.