1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BEHAVIOR OF μ (g, T) FOR T SMALL 15

One could conceivably have the strange situation of a Type Ila singular

solution which forms a Type I singularity model which is a compact shrinking

gradient Ricci soliton. Certainly, one can have a Type Ila singular solution

which forms a Type I singularity model which is a noncompact shrinking

gradient Ricci soliton as evidenced by a degenerate neckpinch which includes

the shrinking round cylinder as one of its singularity models.

MINI-PROBLEM 17.18 (Compact factors of singularity models are shrink-

ers). Show that if the universal cover of a (finite time) singularity model

splits as (Nm, h ( t)) x JR_n-m, where N is compact, then (Nm, h ( t)) is a

shrinking gradient Ricci soliton.

2. Behavior ofμ (g, r) for T small

In this section we present a detailed discussion of the limiting behavior
of the μ-invariant as T tends to 0. As a consequence, we shall show that for
a closed Riemannian manifold on which the isometry group acts transitively,
the minimizer f 7 of W (g, · , T) is not unique for T sufficiently small.

2.1. Behavior ofμ (g, r) for T small.

In the next lemma and proposition we give a belated proof of Lemma

6.33(i), (ii) in Part I regarding the behavior ofμ (g, r) for T sufficiently small

(this is a result of Perelman; see §3.1 of [152]).

LEMMA 17.19 (μ(g,r) is negative for T small). If (Mn,g) is a closed

Riemannian manifold, then there exists f > 0 such that

(17.42) μ (g, r) < 0 for all T E (0, f).

PROOF. Since M is closed, by the short time existence theorem, there

is a f > 0 such that a (unique) solution g (t) to the Ricci flow with g (0) =

g exists for t E [O, f]. Let T (t) ~ f - t and xo E M and consider the

corresponding fundamental solution

u(x,t) ~ (47rr(t))-nl^2 e-f(x,t), x EM, t E [O,f),

to the adjoint heat equation

au

at = -fl 9 ct)u + R 9 ct)u

centered at (xo, f) (i.e., limt)"r u (·, t) = Dx 0 ; note that T (f) = 0).

As Perelman says in §3.1 of [152] and as we have seen in Chapter 16 of

Part II (where we discussed Perelman's differential Harnack estimate v ~ 0),

we have

(17.43) ;~ W (g ( t) , f ( t) , T ( t)) = 0.

Hence, by the monotonicity of the entropy functional,

(17.44)

μ (g, f) = μ (g, T (0)) ~ W (g (0) ,j (0) ,T (0)) ~ ;~ W (g (t) ,f (t) ,T (t)) = 0.