1547671870-The_Ricci_Flow__Chow

16 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

of upper-triangular 3 x 3 matrices

g' ~ { G ~ 0 x, y, z E ~}

endowed with the usual matrix multiplication. Topologically, Q^3 is diffeo-

morphic to IR^3 under the map

Q^3 3ry= f--+(x,y,z)EIR^3.

(~

1 x~ yz

1

)

Under this identification, left multiplication by r corresponds to the map

L'Y(a,b,c) = (a+x, b+y, c+xb+z).

The signature of a Milnor frame on Q^3 is .A = -1 and μ = v = 0. Any

left-invariant metric g on Q^3 may be written in a Milnor frame {Fi} for g as

g = Aw^1 ®w^1 + Bw^2 ®w^2 + Cw^3 ®w^3 ,

where A, B, C are positive. Then the sectional curvatures of ( Q^3 , g) are

A

K (F2 /\ F3) = -3 BC'

and its Ricci tensor is

A

K (F1 /\ F2) = BC'

A^2 A A

Re= 2-w^1 ®w^1 - 2- w^2 ®w^2 - 2-w^3 ®w^3

BC C B

Hence we get a solution ( Q^3 , g ( t)) of the Ricci flow for given initial data

Ao, Bo, Co> 0 if A, B, and C evolve by

d A^2

(1.7a) dt A= - 4 BC

(1.7b) -5._B = 4 A

dt c

(1 7 ). c -5._C dt =^4 B A.

System (1.7) can be solved explicitly. Observing that B/C is a conserved

quantity, we compute that

-5._ (~) = -12 (B) (~)

2

= -12 (Bo) (~)

2

dt B^2 C B^2 Co B2

Solving this gives

A Co/Bo

B^2 12t + BoCo/Ao

and shows that

:tA = -^4 ( ~) (~)A= - 12t + B~Co/Ao A.