1547671870-The_Ricci_Flow__Chow

234 8. DILATIONS OF SINGULARITIES

for all x E Mn. Then since !Rm (t)I = CR, where C > 0 is a constant

depending only on g ( 0), we have

IRm(t)l(T- t)= C

for another positive constant C.
Analogously, if R (x, 0) < 0, then

so T = oo and

n

R(x t)=--~­

, - nR (0)-^1 - 2t'

!Rm (t)I (C1 + t) = C2

for some C1, C2 > 0. Finally if R (x, 0) = 0, then R = 0, T = oo and

!Rm (t)I = C 2: 0.

These considerations lead us to subdivide solutions of the Ricci fl.ow into

two types: those whose curvatures are bounded above by a constant times

the curvature of an Einstein solution with R > 0 or R < 0 (Type I or III)

and those whose curvatures are not (Type Ila or Ilb). More precisely, the

classification of maximal solutions (Mn, g ( t)) by their singularity type is

as follows.

If the singularity time Tis finite, Theorem 6. 45 tells us that the curvature

becomes unbounded,

sup IRml = oo,

Mnx[O,T)

and classify the solution by its curvature blow-up rate - the rate at

which the curvature approaches infinity as time approaches T. (Curvature

blow-up rates were discussed briefly in Subsection 6.2 of Chapter 2.)

DEFINITION 8.1. Let (Mn,g (t)) be a solution of the Ricci fl.ow that

exists up to a maximal time T:::; oo.

• One says (Mn, g ( t)) encounters a Type I Singularity if T < oo
  and
  sup !Rm(-, t)I (T - t) < oo.
  Mnx[O,T)

• One says (Mn, g ( t)) encounters a Type Ila Singularity if T < oo

and

sup IRm (., t)I (T - t) = oo.

Mnx[O,T)

• One says (Mn,g (t)) encounters a Type Ilb Singularity ifT = oo

and

sup !Rm(., t) I t = oo.

Mnx[O,oo)

• One says (Mn, g ( t)) encounters a Type III Singularity if T = oo

and

sup !Rm(-, t)I t < oo.

Mnx[O,oo)