1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6 1. RICCI SOLITONS

is the exponent in the Gaussian function. Here we used the standard identity

£vfgcan = 2'\7'\7 f.

2. Differentiating the.soliton equation - local and global analysis

2.1. Differentiating general solitons. Let (g, X) be a Ricci soliton

structure (gradient or not) on Mn:

(1.17) 2~j + 'ViXj + 'VjXi + Egij = 0.

Condition (1.17) places a strong condition on g and X. For example, con-

tracting (1.17) with g (tracing) gives

(1.18)

. ns

R +div X +

2

= 0,

where div X ~ gii'ViXj and R is the scalar curvature. If Mis closed, then

this implies

(1.19)

2r

E=--,

n

where r ~ JM Rdμ/ vol(M) denotes the average scalar curvature.

Furthermore, we also have the following.

LEMMA 1.10. If (g,X) is a Ricci soliton structure on Mn, then

(1.20)

or more invariantly,

.6.Xb +Re ( Xb) = 0,

where Re : T* M -+ T* M is defined by Re (a) i ~ Rie gem am.

PROOF. Taking the divergence of (1.17) and applying the second con-
tracted Bianchi identity (Vl-3.13) and the Ricci identity (Vl-p. 286b ), we
obtain

=gJ ·k ( 'Vi'VjXk-RjikR.9mXm e ) +.6.Xi

= -'ViR +~elm Xm + .6.Xi,

where we have used (1.18). The lemma follows from cancelling the -'ViR

terms. D

Lastly, computing the scalar curvature of the evolving metric (1.11) and
comparing its time-derivative (at t = 0) with its evolution under the Ricci
fl.ow gives

LEMMA 1.11. If (g, X) is a Ricci soliton structure on Mn, then

(1.21) .6.R + 21Re1^2 = £xR - sR.