222 6. ENTROPY AND NO LOCAL COLLAPSING>.(g(t)) < 0 for some t E [t1, t2], even shrinking breathers, i.e., solutions
with g (t 2 ) = aip*g (t 1 ) and a < 1, are Einstein solutions (trivial Riccisolitons). In order to handle shrinking breathers when >.(g(t)) > 0 for all
t E [t1, t 2 ], we need to generalize the monotonicity formula for the functional
:F to a monotonicity formula for a functional related to shrinking breathers.
This is the monotonicity formula for the entropy W.
In this section we introduce the entropy W and discuss its monotonicity.
We also give a unified treatment of energy and entropy in the last subsection.1.1. The entropy W, its first variation and the gradient flow.
1.1.1. The entropy W. Let 9net denote the space of smooth Riemann-ian metrics on a closed manifold Mn. We define Perelman's entropy func-
tional W: 9net x C^00 (M) x JR+--+ JR by
(6.1) W(g,f,r)~ JMr(R+l\7fl^2 )+f-n-nl^2 e-fdμ
(6.2) =JM [r ( R + l\7 !1
2) + f - n] udμ,
where^3
(6.3)
This is a modification of the energy functional :F (g, f) , which we considered
in the last chapter, where we have now introduced the positive parameter
r. By (5.1), we have(6.4) W(g,f,r) = (47rr)-nl^2 (r:F(g,J)+ JM (f-n)e-fdμ)
(6.5) = (47rr)-nl^2 (r:F (g, J) +N (!)) - n JM udμ,
where the second equality is obtained using definition (5.63). As we shall
see, r plays the dual roles of understanding the geometry of (M,g) at
the distance scale y1T and representing a constant minus time for solutions
(M,g(t)) of the Ricci flow.^4
The functional W has the following elementary properties.
(6.6)(i) (Scale invariance) W is invariant under the scalings r f-7 er and
g f-7 cg, i.e.,W(cg, f, er)= W(g, f, r).
(ii) (Diffeomorphism invariance) If ip : M --+ M is a diffeomorphism,
then
W(g, f, r) = W(ip*g, <p* f, r),
where 'P*g is the pulled-back metric and 'P* f = f o ip,(^3) This u is not to be confused with the u = e-f in Chapter 5.
(^4) We may think of T as physically representing temperature (see §5 of [297]).