238 6. ENTROPY AND NO LOCAL COLLAPSINGNext we prove the existence of a smooth minimizer for (6.49); compare
the proof below with the proof of Lemma 5.22.
LEMMA 6.24 (Existence of a smooth minimizer for W). For any metricg on a closed manifold Mn and r > 0, there exists a smooth minimizer fr
ofW(g,·,r) overX.
PROOF. Again assume r = 1. The lemma will follow from showing that
there is a smooth positive minimizer w 1 for 1i (g, w) under the constraintJM w^2 dμ 9 = 1. A smooth minimizer Ji of W (g, ·, 1) is then given by Ji =
-2logw 1 - ~log (47r) (see Rothaus [312]). We give a sketch of the proof.
Suppose w is such that 1i (g, w) :::::; C 1. Then the above considerations
imply that since JM w^2 dμ 9 = 1,C1~1i (g, w) =JM ( 4 IY'wl^2 + ( R-2logw - ~log (47r) ~ n) w^2 ) dμ
~ 2 JM IY'wl2
dμ - C2,where we used (6.65) below with a= 1. Hence any minimizing sequence for
1-{ (g, ·) is bounded in W^1 >^2 (M). We get a minimizer W1 in W^1 >^2 (M) and
by (6.53), w1 is a weak solution to-4~w1 + Rw1 - 2w1 logw1 - (~ log(47r) + n) w1 = μ(g, l)w1.
By elliptic regularity theory, we have w1 E C^00 (see Gilbarg and Trudinger
[155] for a general treatise on second-order elliptic PDE). Finally, one canprove that w1 > O; see [312] for more details. D
REMARK 6.25. We also have(6.56) μ(g, r) ~inf { W(g, f, r) : f E W^1 '^2 (M), JM udμ = 1},
where C^00 is replaced by W^1 >^2 in (6.49) and u is defined in (6.3).2.3. l\/fonotonicity ofμ. Let (g (t), r (t)), t E [O, T], be a solution
of (6.14) and (6.16) with r (t) > 0. For any to E (0, T], let f (to) be the
minimizer of{W(g (to), f, r (to)) : f E C^00 (M) satisfies (g, f, r) E X}
and solve (6.15) for f (t) backwards in time on [O, to]. By the monotonicity
formula, we have
d
dt W (g ( t) , f ( t) , T ( t)) ~ 0
for t E [O, to]. Note that the integral constraint (6.48) is preserved by the
modified coupled equations (6.14)-(6.16). This can be seen from the follow-
ing calculation:
(6.57)
! JM (47rr)-nl
2
e-f dμ =JM ( ~~ -Ru) dμ =JM (-D*u - ~u) dμ = 0.