- DIFFERENTIATING THE SOLITON EQUATION 7
PROOF. The left-hand side (LHS) is just the usual expression for ~~
under the Ricci fl.ow. To obtain the equality, we compute ~~ by way of the
equality R (T<p*g) = T-^1 <p* (R (g)). Hence ~~ = CxR-cR. DREMARK 1.12. Equation (1.21) can also be derived by taking the di-
vergence of (1.20), commuting derivatives and using the contracted second
Bianchi identity; however this is more tedious.To classify solitons, we must use global techniques like the maximum
principle. The following is in Proposition 5.20 on p. 117 and Lemma 9.15
on p. 271 of Volume One (see Hamilton [186] and one of the authors [218]).PROPOSITION 1.13 (Expanding and steady solitons on closed manifoldsare Einstein). Any expanding or steady Ricci solution (g, X) on a closed
n-dimensional manifold Mn is Einstein. Any shrinking Ricci solution on a
closed n-dimensional manifold has positive scalar curvature.
PROOF. By (1.21),
(1.22) !.lR - (\/ R, X) + 21Re1^2 + cR = 0.
We rewrite this as(1.23) !.l ( R + ~c) - ( ! ( R + ~c) , X) + 21Re+~g1
2- c ( R + ~c) = 0.
At any point xo EM such that R (xo) = Rmin, we have
2 IRc+~gl2- c ( R + ~c) :S 0.
If (g, X) is expanding or steady, then c ;:::: 0. Since R(xo) + 7ke = R(xo) - r
is nonpositive, this implies !Re +~gl^2 (xo) = 0. Tracing then implies
nc
Rmin = R(xo) = -2 = r,and hence R = r = -7ke. Substituting back into (1.23), we conclude that
Re= -~g.
If (g, X) is shrinking, then c < 0. Applying the weak maximum principle
to (1.22), we have R;:::: 0. By the strong maximum principle, either R = 0or R > 0. If R = 0, then (1.22) would imply Re = 0, contradicting the
assumption that (g, X) is shrinking. Hence R > 0. D
We will return to the consideration of shrinking solitons on closed man-
ifolds in Section 7 of this chapter.
2.2. Differentiating gradient solitons. The gradient Ricci soliton
structure equation(1.24)relates the Ricci tensor to the Hessian of f. First note that tracing this gives
nc
(1.25) R + !.lf + 2 = 0.