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We continue our study of $\kappa$-solutions. In the previous lecture we primarily exploited the non-negative curvature of such solutions; in this lecture and the next, we primarily exploit the ancient nature of these solutions, together with the finer analysis of the two scale-invariant monotone quantities we possess (Perelman entropy and Perelman reduced volume) to obtain a important scaling limit of $\kappa$-solutions, the asymptotic gradient shrinking soliton of such a solution.

The main idea here is to exploit what I have called the infinite convergence principle in a previous post: that every bounded monotone sequence converges. In the context of $\kappa$-solutions, we can apply this principle to either of our monotone quantities: the Perelman entropy

$\displaystyle \mu(g(t),\tau) := \inf \{ {\mathcal W}(M,g(t),f,\tau): \int_M (4\pi\tau)^{-d/2} e^{-f}\ d\mu = 1 \}$ (1)

where $\tau := -t$ is the backwards time variable and

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

or the Perelman reduced volume

$\displaystyle \tilde V_{(0,x_0)}(-\tau) := \tau^{-d/2} \int_M e^{-l_{(0,x_0)}(-\tau,x)}\ d\mu(x)$ (3)

where $x_0 \in M$ is a fixed base point. As pointed out in Lecture 11, these quantities are related, and both are non-increasing in $\tau$.

The reduced volume starts off at $(4\pi)^{d/2}$ when $\tau=0$, and so by the infinite convergence principle it approaches some asymptotic limit $0 \leq \tilde V_{(0,x_0)}(-\infty) \leq (4\pi)^{d/2}$ as $\tau \to -\infty$. (We will later see that this limit is strictly between 0 and $(4\pi)^{d/2}$.) On the other hand, the reduced volume is invariant under the scaling

$g^{(\lambda)}(t) := \frac{1}{\lambda^2} g( \lambda^2 t )$, (4)

in the sense that

$\tilde V_{(0,x_0)}^{(\lambda)}(-\tau) = \tilde V_{(0,x_0)}(-\lambda^2 \tau)$. (5)

Thus, as we send $\lambda \to \infty$, the reduced volumes of the rescaled flows $t \mapsto (M, g^{(\lambda)}(t))$ (which are also $\kappa$-solutions) converge pointwise to a constant $\tilde V_{(0,x_0)}(-\infty)$.

Suppose that we could somehow “take a limit” of the flows $t \mapsto (M, g^{(\lambda)}(t))$ (or perhaps a subsequence of such flows) and obtain some limiting flow $t \mapsto (M^{(\infty)}, g^{(\infty)}(t))$. Formally, such a flow would then have a constant reduced volume of $\tilde V_{(0,x_0)}(-\infty)$. On the other hand, the reduced volume is monotone. If we could have a criterion as to when the reduced volume became stationary, we could thus classify all possible limiting flows $t \mapsto (M^{(\infty)}, g^{(\infty)}(t))$, and thus obtain information about the asymptotic behaviour of $\kappa$-solutions (at least along a subsequence of scales going to infinity).

We will carry out this program more formally in the next lecture, in which we define the concept of an asymptotic gradient-shrinking soliton of a $\kappa$-solution.
In this lecture, we content ourselves with a key step in this program, namely to characterise when the Perelman entropy or Perelman reduced volume becomes stationary; this requires us to revisit the theory we have built up in the last few lectures. It turns out that, roughly speaking, this only happens when the solution is a gradient shrinking soliton, thus at any given time $-\tau$ one has an equation of the form $\hbox{Ric} + \hbox{Hess}(f) = \lambda g$ for some $f: M \to {\Bbb R}$ and $\lambda > 0$. Our computations here will be somewhat formal in nature; we will make them more rigorous in the next lecture.

The material here is largely based on Morgan-Tian’s book and the first paper of Perelman. Closely related treatments also appear in the notes of Kleiner-Lott and the paper of Cao-Zhu.

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.