204 5. ENERGY, MONOTONICITY, AND BREATHERS3.1. The infimum A of :F. Suppose we have a steady breather solu-
tion to the Ricci fl.ow with g (t 2 ) = cp*g (t1) for some ti < t2 and diffeo-
morphism cp. One drawback of the energy monotonicity formula is that in
general the solution f to (5.35) has f (t2) -::/= f (t1) o cp, so that in general,
:F (g (t2), f (t2)) -::/= :F (g (t1), f (ti)). By taking the infimum of :F among f,
we obtain an invariant of the Riemannian metric g which avoids this trouble.
DEFINITION 5.21 (.A-invariant). Given a metric g on a closed manifoldMn, we define the functional ,\ : VJtet -t IR by
(5.45) .A(g) ~inf { :F(g, f) : f E C^00 (M), JM e-f dμ = 1}.
Taking w = e-f 12 , we have
(5.46) ..(g) =inf { Q(g, w) : JM w^2 dμ = 1, w > 0},
where, as in (5.5),^9(5.47)Thus, when we fix g and minimize :F (g, f) among f, we are minimizing a
Dirichlet-type functional and we get an eigenfunction-type equation for w.
Aspects of this point of view are discussed in the next two lemmas.
Note that the variation of Q (g, ·) is given by~c5(o,h)g (g, w) = f (4\Jw · \Jh + Rwh) dμ = f (-4/:lw + Rw) hdμ,
JM JM.
where h = c5w. Hence the Euler-Lagrange equation for (note that we dropped
the positivity condition on w)
.A (g) ~inf { Q (g, w) : JM w^2 dμ = 1}
is
(5.48) Lw ~ -4/:lw + Rw = ,\ (g) w.
LEMMA 5.22 (Existence and regularity of minimizer of Q). There exists
a unique minimizer wo (up to a change in sign) of(5.49) inf { Q(g,w): JM w^2 dμ = 1}.
The minimizer wo is positive and smooth. Moreover,(^9) In view of Lemma 5.1(1), the monotonicity of :F exhibits a dichotomy, it is analogous
to both the monotonicty of the total scalar curvature under its gradient flow, ftg =
-2 (Rc-~g), and the monotonicity of the Dirichlet energy under its gradient flow, the
backward heat equation ftw = -b.w. In this sense, the monotonicity of :F exhibits a
beautiful synthesis of geometry and analysis.