1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW 191

1.1. The energy functional F. Let C^00 (M) denote the set of all

smooth functions on a closed manifold Mn. We define the energy func-

tional F: 9J1et x C^00 (M)-+ JR by

(5.1)

Note, in addition to the metric, the introduction of a function f. This embeds

the space of metrics in a larger space. We shall sometimes follow the physics

literature and call f the dilaton.

Since~ (e-f) = (-~f + 1Vf1^2 ) cf, we see from JM~ (e-f) dμ = O

that

(5.2)

So we have two other expressions for the energy:

(5.3) F(g, f) =JM (R + ~J)e-f dμ

(5.4) =JM (R + 2~f - IV fl^2 )e-f dμ.

The second way of expressing the energy is motivated by the pointwise
formula (5.43) in subsection 2.3.2 below.

LEMMA 5.1 (Elementary properties of F).

(1) Dirichlet-type energy. The geometric aspect of F is reflected by

F (g, 0) =JM Rdμ being the total scalar curvature and the function

theory aspect of F is reflected by expressing it as

where w = e-f 12 , which is a Dirichlet energy with a potential term.

(2) Diffeomorphism invariance. For any diffeomorphism 'P of M, we

have

F ( 'P* g, f o 'P) = F (g, f).

(3) Scaling. For any c > 0 and b

F (c2g, f + b) = cn-2e-b:F (g, f).

EXERCISE 5.2. Prove the properties for the energy in the lemma above.

1.2. The first variation of F. We use the symbol o to denote the
variation of a tensor. We shall denote the variations of the metric and

dilaton as og = v E C^00 (T M &Js T M) and of= h E C^00 (M), and we