16 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES
We now rule out μ (g, f) = 0, from which the lemma follows. Suppose
μ (g, f) = 0. Since Wis monotone, we then have
W (g ( t) , f ( t) , r ( t)) = μ (g ( t) , r ( t)) = 0
for all t E [O, f). Hence, by (17.12) we have
(Rc+V'V'f-;r) (t) = 0
fort E [O, f], so that g (t) is a shrinking gradient Ricci soliton with singular
time t = f. In particular,
r (t) max !Rm (g (t))I = const
Mfort E [O, f]. On the other hand, since g (f) is a smooth metric (and recall
that r(f) = 0), we conclude that !Rm(g(t))I = 0 fort E [O,f]. We obtain
a contradiction because there are no fl.at shrinking Ricci solitons on closed
manifolds. D
Recall that Grass's Euclidean logarithmic Sobolev inequality (see Corol-
lary 6.40 in Part I or Theorem 22.15 below) says that if Jo : :!Rn -+ IR is a
smooth function with
then
(17.45)with equality if Jo (x) = lx-;^01
2
for some xo E :!Rn. That is, for Euclidean
space, the entropy is nonnegative and the μ-invariant is zero. Note that
if we let wo ~ (27r )-n/^4 e-fo/^2 , then fJRn w5dμJRn = 1 and we may rewrite
(17.45) as(17.46)
Roughly speaking, since Riemannian manifolds are almost geometrically
Euclidean on small scales, the Euclidean logarithmic Sobolev inequality im-
plies that the entropy on small scales ( r small) is almost nonnegative and
the corresponding μ-invariant is almost zero. The following is in §3.1 of
[152] (see also Proposition 3.2 in Sesum, Tian, and Wang [170]).
PROPOSITION 17.20 (μ(g,r) -+ 0 as r -+ 0). If (Mn,g) is a closed
Riemannian manifold, then
(17.47) lim μ(g,r) = 0.
T-+O+