1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

218 5. ENERGY, MONOTONICITY, AND BREATHERS

4.2.3. The gradient of Hamilton's surface entropy is the matrix Harnack
quantity. A less well-known fact is that the gradient of Hamilton's entropy in
the space of all metrics with the L^2 -metric is the matrix Harnack quantity:

(5.70) OvN (g) =JM Vij (-.:'.1logR · 9ij + \7i\7j logR-~Rgij) dA,

where og = v (see Lemma 10.23 of [111] and use N (g) - E (g) is a con-

stant). In the space of metrics in a fixed conformal class, the gradient is

the trace Harnack quantity. Note that the same relation is true relating

the entropy and the trace Harnack quantity for the Gauss curvature fl.ow of

convex hypersurfaces in Euclidean space [96].

4.3. Bakry-Emery's logarithmic Sobolev-type inequality. The

proofs of Hamilton's surface entropy formula and Perelman's energy formu-

las are formally similar to the proof of Bakry and Emery of their logarithmic

Sobolev-type inequality [18].

PROPOSITION 5.40. Let (Mn,g) be a closed Riemannian manifold with

Re;:::: K for some constant K > 0. If u is a positive function on M, then

JM u log udμ :::; 2 ~ JM u IV log ul

2

dμ +log (Vol ~M) JM udμ) JM udμ.

PROOF. (See [104] for more details of the computations.) Consider the
solution v to the heat equation ~~ = L1v with v (0) = u. The solution v
exists for all time and

t~~ v = Vol ~M) JM udμ.

Define E ( t) -.;.... JM v log vdμ. Then

(5.71) t~~ E (t) =JM udμ ·log (Vol ~M) JM udμ).

We have

ddE =-1 (Vv,Vlogv)dμ=-1 vl\7logvl^2 dμ:::;O.

t M M

Note that limt-+oo ~f (t) = 0.

Using gt logv = .:'.1logv + IVlogvl^2 , we compute

d2E dt { ( )

2 = 2 JM v IVVlogvl

(^2) +Rc(Vlogv, Vlogv) dμ.
Using our assumption Re;:::: K, we find
d^2 E dE
->-2K-. dt (^2) - dt