1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

196 5. ENERGY, MONOTONICITY, AND BREATHERS

LEMMA 5.8 (Modified contracted second Bianchi identity).

(5.21)

1.4. The functional ;::m and its gradient flow. Unlike F (g, J) , we

can obtain a functional of just the metric g by fixing a measure dm on a

closed manifold Mn; by a measure we mean a positive n-form on M.^6

Define ;::m : 9.net -+ JR by

(5.22)

where

(5.23) f-:-. log (dμ) dm.

REMARK 5.9. The expression (5.23) makes sense because, given a fixed

measure dm on Mn, we can define the bijection

C^00 (A nr* M) -+ C^00 (M) ,

w f-+ <p,

where <p is defined so that w = <pdm (here we have used the fact that

Anr;M ~JR). Thanks to this, it is possible to define the quotient of two

n-forms; e.g., if w1 = <p1dm and w2 = <p2dm, where 0, then we set

W1 _,__ <p1

W2 <p2

Without using the notation f, we can write the energy of the metric g

as

Using the modified Ricci and scalar curvatures, we can rewrite

?(g) = JMijRfjdm= JM Rmdm.

REMARK 5.10. Let <p : M -+ M be a diffeomorphism. Note that in

general

r (<p*g) -=1= r (g).

That is, by fixing the measure dm, WE? get P (g) , which breaks the diffeo-

morphism invariance of F (g, f). In subsection 3.1 of this chapter we shall

solve this problem by considering a functional A (g) which is diffeomorphism-
invariant.

(^6) For a calculational motivation for fixing the measure, see the notes and commentary
at the end of this chapter.