236 6. ENTROPY AND NO LOCAL COLLAPSINGDEFINITION 6.20 (Infimum invariantsμ (g, T) and v (g)). The function-
alsμ : 9J1et x JR+ ->JR and v : 9J1et-> JR are defined by(6.49) μ(g, T) ~inf {W(g, f, T) : f E C^00 (M) satisfies (g, f, T) EX}'
(6.50) v(g) ~inf {μ(g,T): TE JR+}.Note that we do not assume that the two functionals μ(g, T) and v(g)
take finite values. We will see later that μ(g,T) is always finite for anygiven g and T, and in the important case where >.(g) > 0 (where the proof
of the nonexistence of nontrivial expanding breathers cannot be applied to
the shrinking ones), we have that v(g) is finite. We have the following
elementary properties of these two functionals.
(i) (Continuous dependence of μ on g and T) μ(g 8 , T) is a continuous
function of ( S, T) for any C^2 family g 8 • lO
(ii) (Continuous dependence of v on g) v(g 8 ) is a continuous function
for any C^2 family gs.
(iii) (Scale invariance) It follows from the scaling property of W that
we have
μ(g, T) =μ(cg, CT),
v(g) = v(cg).
(iv) ( Diffeomorphism invariance) Since W is invariant under a diffeo-morphism : M -> M, we have
μ(g, T) = μ (<l>*g, T),
v(g) = v(<I>*g).
Compare (i) and (ii) with Lemma 5.24.
EXERCISE 6.21. Prove properties (i)-(iv) above.Using the fact that the variation of W : X -> JR with respect to f is
O(o,h,O) W (g, f, T) =~JM Th ( 2.6.f + f - (; + l) - IV fl^2 + R) udμ,
where h satisfies JM hudμ = 0, we have the Euler-Lagrange equation of
(6.49),
T ( 2.6.f - IVJl^2 + R) + f - n = Cfor some constant C. If f 7 is a minimizer of (6.49) (we will see the existence
of JT in the next subsection), then it follows that
μ(g,T) =JM [T(2.6.JT-l\7JTl^2 +R) +JT-n] (47rT)-nl^2 e-lrdμ,
and hence C = μ (g, T) for a minimizer. Therefore we have the following.
(^10) It is easy to see that μ(g, T) is semi-continuous in g. This is a consequence of the
fact that if a function h(x, y) is continuous, then infyEY h(x, y) is upper semi-continuous
in x.