1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. DIFFERENTIATING THE SOLITON EQUATION 9

In conclusion, the new identities we obtain by differentiating f up to

fourth order are

(1.29)

(1.30)

-ViRjk + VjRik = -Rijk.e.\Jd,

ViVjRke - ViVkRje = ViRjk£m Vmf - Rjk£m (Rim+ ~9im)

and the traces are

(1.31)

(1.32)

(1.33)

where

1

-V·R 2 i = R-nVnf t<. {. '

Mke = ( -V eRkm + V mRke) V mf,

!J.R + 2 IRcl^2 +ER= V f · V R,

1 c

Mke ...;.. /J.Rke - 2 V kV eR + 2Rik£mRim - RkmRme + 2 Rke.

The last equation, which is the trace of the equation above it, is a special

case of (1.21).

REMARK 1.14. The quantities in (1.29) and (1.32) appear in the matrix
Harnack quadratic discussed in Part II of this volume (see also subsection
4.2 of Appendix A).

We will finish this subsection with an application of Lemma 1.10 to
gradient solitons. Substituting X = V f in (1.20) yields

!J.(Vd) + Riegemv mf = 0.

On the other hand, commuting covariant derivatives gives

!J.(Vd) = Vi(!J.f) + Rik9k£Vd.

Combining these equations with (1.25) and (1.9) yields

0 = ViR + 2l£ViVkfVd + cVd = Vi(R +IV fl^2 +cf),

proving the following.

PROPOSITION 1.15 (Constant gradient quantity on solitons). If (g, V f, c)

is a gradient Ricci soliton structure on a manifold Mn, then

(1.34) R + IV f 12 +cf = const

is constant in space. Consequently, by (1.25),

(1.35) R + 2/J.f - IV f 12 - cf = const.

Formula (1.34) is used in the study of the geometric properties of gra-

dient Ricci solitons; see Chapter 9 of [111] for an exposition. A significance

of the quantity on the LHS of (1.35) will be exhibited in the use of (5.42)

and (5.43) to prove energy monotonicity in Chapter 5.

With additional hypotheses, we can determine the constant in the propo-

sition above (see also Theorem 20.1 in [186]).