1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

44 1. RICCI SOLITONS

Because of the fl.at factor, R^2 = 2JRcJ^2 , and therefore dr/dt = r^2 /3. This

implies that r increases without bound, which is impossible for a soliton or

a breather. 0

A similar (but more elaborate) argument based on a pinching set can

be used to prove that a nontrivial soliton for the normalized Ricci fl.ow on a

compact Kahler surface must have curvature at least as negative as Koiso's

example; see [222].

8. Perelman's energy and entropy in relation to Ricci solitons

The notion of gradient Ricci soliton has motivated the discovery of mono-

tonicity formulas for the Ricci fl.ow, which in turn have useful geometric ap-

plications. Here we consider some monotone integral quantities. In particu-

lar, in Chapter 5 we shall further study Perelman's energy functional:

(1.68)

where (Mn, g) is a closed Riemannian manifold and f : M ----+ R As we will

see in (5.31) and (5.41), this functional is nondecreasing under the following

set of evolution equations (see below for a motivation for considering (1.70)):

(1.69)

(1.70)

09ij {)t -- -2R-. iJ,

of 2

at = -~f +IV fl -R.

In particular, we have the following.

THEOREM 1.72 (Energy monotonicity). For any solution (g (t), f (t)) of

(1.69)-(1.70) on a closed manifold Mn, we have

!F(g(t), f (t)) = 2 JM IRij + \7i\7jfl^2 e-f dμ 2': 0.

Hence -itF(g(t), f (t)) = 0 at some time to if and only if

(1.71) Rij + \7i\7jf = 0

at time to. That is, g (to) is a steady gradient Ricci soliton fl.owing along

\7 f (to).

We can motivate the consideration of equations (1.69)-(1.70) by seeing

how it relates to a steady soliton g fl.owing along a gradient vector field \7 f

and in canonical form. By (1.14) and (1.25),

~~ = 1Vfl

2

= 1Vfl

2

-R-~f,

which is (1.70).
A similar consideration for shrinking gradient solitons can be used to

motivate the study of Perelman's entropy functional W (g, f, T) discussed in