1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

280 6. ENTROPY AND NO LOCAL COLLAPSING

REMARK 6.92. Note that

(V'i - V'd) ('Yj - Y'jf) Vij

=div (divv) +\Re, v) - 2 div (v) · V' f + VijV'dV'jf - (Rij + Y'iV'jf) Vij·

Also,

and

For any closed Riemannian manifold (.Mn,§) with variation gs§ = v,

by (5.10), (5.51), and JM(~ - h) e-f dμ = 0, we have

:s >.. (§) = - JM Vij (Rij + V'iV'jf) e-f dμ,

where f is the minimizer of :F (§, ·). Thus the critical points of ).. (§) are the

steady gradient Ricci solitons fl.owing along the gradient of the minimizer.

Since, given v, equality holds in (6.112) when h satisfies (6.111), we obtain

the following second variation formula proved in [53].

THEOREM 6.93 (Cao, Hamilton, and Ilmanen). Let (.Mn,§) be a closed

Ricci fiat manifold and let v be a symmetric 2-tensor. If g ( s) = §+sh, then

d

2

ds2^1 s=O).. (g (s)) =JM r ( -2^1 IV'vl^2 + jd1vvl.^2 -^1 2 IV'wj^2 + ~pjqViqVjp ) dμ

= JM \L v, v) dμ,

where w is defined by (up to an additive constant)

l:::..w ~div (divv)

and

Lv ~ ~l:::..v - ~.L(divv)Q§ +Rm (v).

Similarly, one can compute the variation of

v (§) ~ inf μ (§, T) = inf W (§, f, T) ,

TE[O,oo) 7',/

where JM e-f dμ = (47rrr/^2 , on an Einstein manifold with positive scalar
curvature (see [53]).

THEOR8M 6.94 (Cao, Hamilton, and Ilmanen). Let (.Mn,§) be a closed

Einstein manifold with positive scalar curvature, i.e., Re= 2 ~§ where T > 0,