1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

10 1. RICCI SOLITONS

COROLLARY 1.16. If (g, \7 f) is a steady gradient Ricci soliton structure

on Mn with positive Ricci curvature and if the scalar curvature attains its

maximum at a point 0, then

R+ J\7fJ^2 = R(O).

PROOF. We have Rij = -\7i\7jf > 0 and by Proposition 1.15, R+J\7fJ^2

is constant. On the other hand, at 0 we have 0 = \7 i J \7 f J^2 = -2Rij g^1 "k \7 k f,

which implies that \7 f ( 0) = 0. D

REMARK 1.17. If we only assume the Ricci curvature is nonnegative,

then there is a counterexample by simply taking the fiat metric on the xy-

plane and letting f be the x-coordinate, in which case Rj = - \7 i \7 j f = 0,

so that R+ J\7JJ^2 = const > R(O) = 0.

Quantities which are constant on gradient Ricci solitons are useful in the

study of their geometries; see Chapter 9 of [111] for an exposition of some

examples of this.

2.3. Ricci soliton structures and exterior differential systems.

We end this section with a local analysis of Ricci soliton structures. From

(1.17), which defines a Ricci soliton structure (g, X), one might hope to

obtain more stringent tensorial local conditions on g and/ or X by a combi-

nation of differentiation, contracting, and equating mixed partials. However,

no further lower-order (i.e., first-order in X and second-ordering) identities

arise this way. This follows from applying the machinery of exterior differ-

ential systems to the soliton equation (1.17), written in local coordinates

as a system of PDE that is second-order in the entries of g and first-order

in the components of X. The theory of exterior differential systems - in

particular, the Cartan-Kahler Theorem^3 - is able to predict when a

system of PDE has solutions, and how large the solution space is, provided

the system passes a test indicating that it is involutive.^4 In addition, if

the system is involutive, then any k-jet of a solution can be extended to a

(k + 1)-jet of a solution, and so on to a convergent power series solution.

In the case of (1.17), in order to obtain an involutive system, one has

to reduce the enormous size of the solution space (which is due to the dif-

feomorphism invariance of the soliton condition) by adding the requirement

that the local coordinates are harmonic functions with respect to the metric

g. Since D. = gij ( aiaj - I'fjok) , this is a first-order condition on the entries

of g and takes the form

(1.36) ljrfj = o,

where I'fj are the Christoffel symbols. This is just another variation on

DeTurck's trick for proving short-time existence for the Ricci fl.ow (see

(^3) See Theorem III.2.2 and Corollary III.2.3 on pp. 80-88 in [36] or Theorem 7.3.3 on
pp. 254-256 in [223].
(^4) See Chapter III, pp. 58-65 of [36] or Chapter 7, p. 256 in [223].