1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

sa 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Therefore mg(-l) (E 1 ;i) = 0 for each i E N. Since Ea

mg(-l)(Ea) = 0. We conclude that m 952 (Ea)= O; that is,

LJE1/ii we obtain

iEN

(29.41) R 00 = 0 a.e. on 52.

We remark that by the strong maximum principle and Klingenberg's t heorem

(see (29.69) below), we see that for a ny xa E 52 - Ea and any ti --+ -oo, the

solutions ( 52 , g ( t +ti) , xa) subconverge in the C^00 Cheeger- Gromov sense to a

complete flat solution, which must b e IR^2 or a cylinder. This is consistent with the

conclusions of Theorem 29.2. Later , we shall show that R 00 = 0 on 52 except at a

pair of points; see Proposition 29.36 below.

5.2. Proof using an energy monotonicity formula.

To understand the b ackward limits of g (t) better (i. e., working towards their
classification), it is useful to consider monotonicity formulas for quantities associ-
ated to g(t). In the Type I case, entropy monotonicity has previously b ee n applied.
In this subsection we consider a 1-parameter family of monotone functionals which
generalize Polyakov's energy functional for the Ricci flow on surfaces. The mono-

tonicity of one of these functionals gives another proof that R 00 = 0 a.e. on 52.

Define a family 113 of Dirichlet-type energy functionals on the space of Rie-

mannian metrics g = tg 52 on 52 by

(29.42) 113 (g) ~ fs

2 c\7:J~

2

+ F 13 (ii)) dμ52,

where F 13 (ii) = - 2 ~ 13 ii^2 -13 for f3 =f. 2 and where F 2 (ii) = -4 ln ii. We compute for

the ancient solution g(t) that

(29.43) -1d 13 (g (t)) = j ( -(3--i'Vvl~^2 - 2d1v. (\l- v) - 4v l-f3) av -dμ5 2

dt 52 vl3+l vl3 at

= .l 2 ( -2 ~~ - (2 - (3) 1vv1~2) JL dμ52,

which is nonpositive whenever f3 :::; 2.

We note two important special cases. First, the functional

(29.44)

is equivalent to Polya kov's energy, i.e., the logarithm of the determinant of the

Laplacian of g. Indeed, a standard formula (see (A.17) in [77] for example) yields

1

log det 6. 9 - log det 6. 52 = --1 2 (g) + ln Area(g) - ln( 47f).

487f

In this case the RHS of (29.43) can be rewritten as -2 J 52 R^2 dμ 9 (see (B.24) in [77]

for example). In the space 9Jlet of metrics in a conformal class on 52 , the Ricci flow
is the negative gradient flow of 12 with res pect to the L^2 -metric on 9Jlet.
Second, for the present purposes, Daskalopoulos, Hamilton, and Sesum consider
the functional

(29.45) l1(g(t))= { (~l'Vvl~2-4v)dμ52=- { (R+2v)dμ52.

} 52 v } 52