1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

462 A. BASIC RICCI FLOW THEORY

THEOREM A.27 (RF on closed 3-orbifolds with Re> 0). If (V^3 ,go) is

a closed Riemannian 3-orbifold of positive Ricci curvature, then a unique

solution g (t) of the normalized Ricci flow with g (0) = go exists for all

t > 0, and as t--+ oo, the g(t) converge to a metric g 00 of constant positive

sectional curvature. In particular, V^3 is dijf eomorphic to the quotient of 83

by a finite group of isometries.

One of the main ideas in the proof of Theorem A.26 is to apply Theorem

A.25 to obtain pointwise curvature estimates which lead to the curvature

tending to constant as the solution evolves. From now on we shall assume

that ,\ (t) 2: μ (t) ~ v (t) are solutions of the ODE system (Vl-6.32). The

evolutions of various quantities and their applications to the Ricci fl.ow on

closed 3-manifolds via the maximum principle for systems are given as fol-

lows.

(1)

(A.20)

(2)

(A.21)

(A.22)

with the inequality holding whenever μ + v ~ 0. So Re > 0 is

preserved in dimension 3 under the Ricci fl.ow.

.!!:_log (-,\-) = μ2 (v - ,\) + v2 (μ - ,\) < 0.

dt v + μ ,\ (v + μ) -

If go has positive Ricci curvature, then so does g (t) and there exists

a constant C1 < oo such that

,\(Rm) :S C1 [v (Rm) +μ(Rm)].

(3) If v + μ > 0, then

-log d ( --v )

dt (v + μ + ,)1-8

= 6 (v + ,\ - μ) - (1 - 6) (v + μ) μ + (μ - v) ,\ + μ

2

v+μ+-\

2

:S 6 (v + ,\ -μ) - (1 - 6) μ

v+μ+,\

Since v + ,\ - μ :S ,\ :S 2C1μ and v+~+>-2: vtf' ~ 6 b 1 , choosing 6 > 0 small

enough so that 1 ~ 8 :::; 12 ~ 2 , we have
1

-log d ( ,\-v ) :S 0.

dt (v + μ + ,\)1-8