- PERELMAN'S ENERGY AND ENTROPY IN RELATION TO RICCI SOLITONS 45
Chapter 6 and the ·associated equations for g, f and 7 > 0. In particular,
analogous to F we define in (6.1) the entropy:W(g, f, 7) ~JM [7 ( R +IV' f1^2 ) + f - n] (41f7)-nl^2 e-f dμ.
Under the system of equations
0
(1.72) ot9ij = -2Rij,
of 2 n
(1.73) 8t = -b.f + IY'fl -R+ 27'(1.74)d7
-=-1
dt 'we shall show in ( 6.17) the following.
THEOREM 1.73 (Entropy monotonicity). If (g (t), f (t), 7(t)), 7 (t) > 0,
is a solution of (1.72)-(1.74) on a closed manifold Mn, then
d
dt W(g(t), f (t), 7(t))=JM 27 /Rij + \7i\7jf-~; /
2(41f7)-nf^2 e-f dμ 2: 0.
Note that the right-hand side (RHS) vanishes, i.e., ft W(g(t), f(t), 7(t)) =
0, if and only if g (t) is a shrinking gradient Ricci soliton. The above mono-
tonicity formulas beautifully display the utility of considering Ricci solitons.
REMARK 1. 7 4. Assuming that (Mn, g ( t)) is a shrinking gradient Riccisoliton in canonical form (1.13) with.\= -1, we have
Re (g (t)) + \7^9 (t)\7g(t) f (t) -
2~g (t) = 0,
where ~; = -1, and hence (1.14) implies f satisfies (1.73).
Finally we consider the expander entropy of Feldman, Ilmanen, and one
of the authors [143]. Define the functional
W+(g,f+,7)~ JM [7(R+IY'f+l^2 )-f++n] (41f7)-nl^2 e-f+dμ.
Under the system
(1.75)
(1.76)
(1.77)
0.
ot9ij = -2Rij,
of+ 2 n
at = -b.f+ +IV' f+I - R-27'd7 = l
dt 'we have the following.