1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

206 5. ENERGY, MONOTONICITY, AND BREATHERS

Finally we show wo > 0. By the Hopf boundary point lemma (see Lemma

3.4 of Gilbarg and Trudinger [155]), if wo = 0 somewhere, then there exists
a point Xo E an, where n = {x EM : Wo (x) > O}' such that an satisfies
the interior sphere condition at xo, so that w (xo) = 0 and j\7w (xo)I i= 0,
which is a contradiction to wo 2: 0.
Finally, properties (1) and (2) follow easily. D
The existence of a unique positive smooth minimizer wo of g(g, w) un-
der the constraint JM w^2 dμ = 1 implies the existence of a unique smooth

minimizer Jo of :F(g, ·) under the constraint JM e-f dμ = 1. From (5.50) we

see the following.

LEMMA 5.23 (Euler-Lagrange equation for minimizer of F). The mini-

mizer f o = -2 log wo of F (g, ·) is unique, C^00 , and a solution to

(5.51) .\ (g) = 2!::.fo - l\7fol^2 + R.

That is, the modified scalar curvature is a constant, i.e., Rm = .\ (g).

Note that from setting v = 0 in (5.10), for the minimizer f of (5.45), we
have

6(0,h)F (g, f) = - JM h ( 2!::.f - j\7 fl^2 + R) e-f dμ

for all h such that JM he-! dμ 9 = 0. We can also obtain (5.51) directly from

this.
We summarize the properties of the functional .\ on a closed manifold
Mn.

(i) (Lower bound for.\) .\(g) is well defined (i.e., finite) since

F(g, f) 2: min R (x) · [ e-f dμ =min R (x) ~ Rmin·

xEM }M xEM

In particular,

.\ (g) 2: Rmin·

(ii) (Diffeomorphism invariance) If <p : M ---+ M is a diffeomorphism,

then

.\(<p*g) = .\(g).

(iii) (Existence of a smooth minimizer) There exists f E C^00 (M) with

JM e-f dμ = 1 such that .(g) = :F(g, J), i.e.,

(5.52) .(g) =JM (R + j\7 fj^2 )e-f dμ.

(iv) (Upper bound for .\) We have

(5.53) .(g) :::; Vol~M) JM Rdμ.

This can be seen by choosing f =log Vol(M), which satisfies

JM e-fdμ 9 =1 and .(g):::; JM(R+ [\7f[^2 )e-f dμ.