1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

456 A. BASIC RICCI FLOW THEORY

(xo, to). However, since d(·, q) is 02 at xo, we may apply the maximum

principle to the equation for G. One checks (Exercise: Prove this) that

G (x, t)::::; O~

on M x [O, T), where O~ ----+ 02 as 6----+ 0. Hence

max_ (t¢F) ::::; 02.

Mx[O,t]

2. Basic Ricci fl.ow

The (unnormalized) Ricci flow equation on a manifold Mn is

a

(A.18) ot9ij = -2~j,

whereas its cousin, the (volume-preserving) normalized Ricci flow equation

on a closed manifold, is

(A.19)

a 2r

ot 9ij = -2Rij + -:;;: 9ij,

where r (t) ~ (JM Rdμ/ JM dμ) (t) is the average scalar curvature.

REMARK A.12. The variation of a Riemannian metric in the direction of

the Ricci curvature was considered by Bourguignon (see Proposition VIII.4

of [32]). A fundamental work using a nonlinear heat-type equation (the

harmonic map heat flow) is by Eells and Sampson [135].

Substituting h = -2 Re into Lemma A.l yields the following result,

which gives the evolution equations for the Levi-Civita connection and cur-

vatures under the Ricci flow (A.18). (See Corollary 6.6 (1) on p. 175,

Lemma 6.15 on p. 179, and Lemmas 6.9 and 6. 7 on p. 176 of Volume One.)

COROLLARY A.13 (Ricci flow evolutions). Suppose g (t) is a solution of

the Ricci flow: gtg = -2 Re.

(1) The Levi-Civita connection r of g evolves by

{) k kC

(Vl-6.1) ot I'ij = -g ('\JiRjc + 'ljRic - 'lcRij).

(2) Under the Ricci flow, the (4, 0)-Riemann curvature tensor evolves

by

(Vl-6.17)

a

at Rijkc = 6.Rijkc + 2 (Bijkc - Bijck + Bikjc - Bicjk)

• (Rf RpjkC + R~RipkC + R1RijpC + R~Rijkp),

where

(Vl-6.16) B-iJ 'k"--'--" -;--gPrgqsR· ipJq. Rk r<es n -- -Rq pij RP q/!,k"