1547671870-The_Ricci_Flow__Chow

18 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

so the Ricci flow is equivalent to the system

(1.lOa)

d C^2 - A^2

dtA =^4 BC

(1.lOb) ~B=4(A+C)2

dt AC

(1.lOc)

d A^2 - C^2

dtC =^4 AB

Observing that AC = AoCo and B (C - A) = Bo (Co - Ao) are conserved

quantities, we define G ~ A/C and consider the simplified system

(1.lla) :tB = 8 + 41 +GG2

d 1 - G^2

( 1. 11 b) dt G = 8 B ,

which has a solution for all positive time. Observing that ftB 2: 16, we

conclude in particular that B ___. oo as t ___. oo. If Ao = Co, then G = 1

and B is a linear function of time. If not, then it is straightforward to show

that G is strictly monotone and approaches a limit G 00 satisfying either

Go < Goo :S 1 or 1 :S G 00 < Go. Since

d 1 1 + G

dG log B = 2G 1 - G '

the condition G 00 -/= 1 is incompatible with the observation that B ___. oo. It

follows easily that A, C ___. J AoCo and that B grows linearly.

PROPOSITION 1.23. For any choice of initial data Ao, Bo, Co > 0, the

unique solution of (1.10} exists for all positive time. For any c > 0, there

exists TE 2: 0 such that

and

l

~dt B- 161 < - c

for all t 2: TE. Moreover, there exist constants 0 < c1 :::; c2 < oo depending

only on the initial data such that each sectional curvature K is bounded for

all t 2: 0 by

C1 < K < C2.

t - - t

COROLLARY 1.24. On any compact sol-geometry manifold, the normal-

ized Ricci flow undergoes collapse, exhibiting Gromov- Hausdorff convergence

to JR.

To gain insight into the behavior of sol metrics on compact manifolds, we

recall a construction from [ 67 ]. One chooses A ESL (2, Z) with eigenvalues

A+ > 1 > .._. Then in coordinates e, x, y on JR^3 , chosen so that the x, y axes

coincide with the eigenvectors of A, one defines an initial metric by

(1.12) g ~ e^2 a. de® de+ ef3+-y dx ® dx + ef3--y dy ® dy,