26 27. NONCOMPACT GRADIENT RICCI SOLITONSIn general, for shrinkers, although limr-+oo Vol~no(r) exists for each 0 EM and
is bounded above only depending on n, we do not know if Vol~no(r) is bounded
above independent of 6 E M and r ;::: 1. In particular, does there exist C < oo
such that VolB 0 (1):::;: C for all OEM?5. Logarithmic Sobolev inequality
In this section we discuss the logarithmic Sobolev inequality on shrinkers due
to Carrillo and one of the authors, which is based on the works of Bakry and Emery,Villani, and others. Let Q = (Mn, g, f, -1) be a complete noncompact shrinking
G RS structure.Taking T = 1 in Perelman's entropy functional, we define the (scale 1) entropy
as(27.115) W(g, c.p) ~ W(g, c.p, 1) =JM (R + IVc.pl^2 + c.p - n)(47r)-nf^2 e-'Pdμ.
Define the μ-invariant (or logarithmic Sobolev constant) of g by
μ(g) ~inf W(g, c.p),
cp
where the infimum is taken over all c.p : M -+ IR U { oo} such that w ~ e-cp/^2 E
C~(M) satisfies the constraint JM w^2 dμ = (47rt1^2.
Now assume that f satisfies the normalization JM e-f dμ = (47rt 12. Then for
some constant C 1 (Q) we have
(27.116)
Define the entropy of Q to be μ(Q) ~ W(g, f). In view of the exponentially
decaying e-f factor and the volume bound (27.106), one can prove using the es-
timates for f, IV f 12 , and D.f in §2 of this chapter that the integral defining μ(Q)
converges and that we have the equality
JM IV !1
2
e-f dμ =JM D.f e-f dμ.Hence
(27.117)By (27.2 ) and (27.116), we have
(27.118) R + 2D.f - IV fl^2 + f - n = f - R - IV fl^2 = -C 1 (Q).
Therefore, by (27.117) we have that μ(Q) = -C 1 (Q).
Carrillo and one of the authors proved that μ(g) = μ(Q), that is,
THEOREM 27.46 (Sharp logarithmic Sobolev inequality for shrinkers). If Q =
(Mn, g, f, -1) is a complete noncompact shrinking GRS, then
infW(g,c.p) cp = μ(Q) = -C1(Q),where the infimiim is taken over all c.p : M -+ IR U { oo} such that w ~ c'P/^2 E
C~(M) satisfies the constraint JM w^2 dμ = (47rt/^2.