- A GEOMETRY WITH TRIVIAL ISOTROPY 17
Observing that AB is another conserved quantity, we can thus obtain the
full solution
(1.8a)(1.8b)(1.8c)A= A~^13 B6^13 C6^13 (12t + BoCo/Ao)-^1!^3
B = A~^13 B~^13 C0^113 (12t + B0Co/Ao)^1!^3
C = A~/^3 B 0113 c~l^3 (12t + B0Co/Ao)^1!^3.
Thus we have proved the following.PROPOSITION 1.21. For any choice of initial data Ao, Bo, Co > 0, the
unique solution of (1. 1) is given by (1.8). There exist constants 0 < c 1 :S
c2 < oo depending only on the initial data such that each sectional curvature
K is bounded for all t ;:::: 0 byC1 < K < C2
t - - t)
and such that the diameter of any compact quotient N^3 of Q^3 is bounded for
all t;:::: 0 byc 1 t^116 < - diamN^3 < - c 2 t^116.
In particular, N^3 is almost fiat.Recall that a Riemannian manifold (Mn,g) is said to be c:-fiat if it£
curvature is bounded in terms of its diameter by
c:
(1.9) IRml <.- diam^2 (Mn, g)
One says Mn is almost flat if it is c:-fiat for all c: > 0.
COROLLARY 1.22. On any compact nil-geometry manifold, the normal-
ized Ricci flow undergoes collapse, exhibiting Gromov-H ausdor.fJ convergence
to (~^2 , 9can) , where 9can is the standard fiat metric.7. The Ricci flow of a geometry with trivial isotropy
The only geometry with trivial isotropy is sol, which may also be re-
garded as the group Isom --------(lE~) of rigid motions of Minkowski 2-space.The signature of a Milnor frame on sol is A = -1, μ = 0, and v = 1.
Any left-invariant metric g may be written in a Milnor frame {Fi} for gas
g = Aw^1 ®w^1 + Bw^2 ®w^2 + Cw^3 @w^3.
The sectional curvatures are
(A- C)^2 - 4A^2
K (F2 /\ F3) = ABC(A+ C)^2
K (F3 /\Fi)= ABC(A-C)^2 - 4C^2