1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

322 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

2.3. The Harnack quantity.
We proceed to give the proofs of Theorem 25.8 and Theorem 25.9. Let
u be a positive solution to the heat-type equation (25.52) and let

(25.62) L ~ logu.

Note that (25.52) implies

(25.63) ~~ = !J.L + l\7 £1^2 - Q.

Here and below, the inner products and norms are with respect tog (T).

Given E E JR., define the gradient-type (Harnack) quantity

oL

(25.64) P ~ OT - (1-c) l\7 £1^2 + Q = /J.L + E l\7 £1

2

.

(Later, we shall choose E positive and small enough.)

REMARK 25.16.

(1) In the model Euclidean case, where u is the fundamental solution

to the heat equation on JR.n (Rij = 0) and E = 0, we have that

rP = T~L = T~x ex ~YI') = -~

is constant.

(2) For the heat equation on a static manifold with Re~ 0 one consid-

ers the quantity P with E = 0 and shows that P ~ -;;_.

We wish to compute the evolution of P. Since /:::;. = gij ( OiOj - I'fjok),

by (25.49) and Lemma 3.2 in Volume One, we have

(25.65) OT^0 ( !J.g(r) ) = -2Rij\7i\7j - g i' J (\7iRjfl. + \7jRifl. - \7gRij) g kfl. \71,,

= -2Rij\7i\7j + (-2lj\7iRjk + \7kR) · \7k

as differential operators acting on functions.

Recall that in the special case where Rij = cRij, c E JR., is a constant

multiple of the Ricci tensor, by the contracted second Bianchi identity, the

evolution of the Laplacian simplifies to

0

(25.66) OT (!J.g(r)) = -2C~j\7i\7j.

2.4. Evolution of the Harnack quantity.

We have the following equation (compare with (E.39) and (E.40) in Part

II).

LEMMA 25.17 (Evolution equation for the Harnack quantity). If u is a

positive solution to the heat-type equation (25.52), where the time-dependent

metric g (T) satisfies (25.49), then for any EE JR. the Harnack quantity

p = /J.L + E J\7 LJ

2

'