1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

464 A. BASIC RICCI FLOW THEORY

then

(A.24) I ( )I

C (n, K, r, a) _ C(n, JKr, a)K

V'Rm y,t :::; Vt - Vt

for all (y,t) E Bg(O) (p,r/2) x (0,T]. Given in addition (3 > 0 and/> 0,

if also / / VK ::=:; r ::=:; (3 / JK, then there exists C ( n, a, (3, /) such that under

the above assumptions

in Bg(O) (p,r/2) x (0,T].

K

IY'Rml ::=:; C(n,a,(3,1) Vt

For a proof and applications, see W.-X. Shi [329], [330], Hamilton [186],

[111], or Part II of this volume.

2.6. The Hamilton-Ivey estimate. The following result reveals the

precise sense in which all sectional curvatures of a complete 3-manifold evolv-
ing by the Ricci fl.ow are dominated by the positive sectional curvatures. (See
[186] or Theorem 9.4 on p. 258 of Volume One.)

THEOREM A.31 (3d Hamilton-Ivey curvature estimate). Let (M^3 , g ( t))

be any solution of the Ricci flow on a closed 3-manifold for 0 ::=:; t < T.

Let v (x, t) denote the smallest eigenvalue of the curvature operator. If

infxEM v (x, 0) 2:: -1 1 then at any point (x, t) EM x [O, T) where v (x, t) < 0,

the scalar curvature is estimated by

(A.25) R 2:: lvl (log lvl +log (1 + t) - 3).

2. 
   
   
   
   7. Ricci solitons. If g is a Ricci soliton, then
   
   
   
   

Re _f!_g = Cx~g

n
for some p E IR and 1-form X. Under this equation we have

(Vl-5.16) .6. (R - p) + (\7 (R-p), X) + 2 jRc-~gl

2

+ ~ (R - p) = 0.

Using this formula, in Proposition 5.20 on p. 117 of Volume One, the fol-
lowing classification result for Ricci solitons was proved. (See Chapter 1 of
this volume for the relevant definitions.)

PROPOSITION A.32 (Expanders or steadies on closed manifolds are Ein-

stein). Any expanding or steady Ricci soliton on a closed n-dimensional

manifold is Einstein. A shrinking Ricci soliton on a closed n-dimensional

manifold has positive scalar curvature.

In dimension 2, all solitons have constant curvature. (See Proposi-
tion 5.21 on p. 118 of Volume One.)

PROPOSITION A.33 (Ricci solitons on closed surfaces are trivial). If

(M^2 ,g(t)) is a self-similar solution of the normalized Ricci flow on a Rie-

mannian surface, then g(t) = g(O) is a metric of constant curvature.