1547671870-The_Ricci_Flow__Chow

10. DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 145

COROLLARY 5.56. Q satisfies the evolutionary inequality

a

(5.38) at Q 2 b.Q + 2 (VQ, \7 L) + Q^2 + rQ.

PROOF. Use (5.37) and the standard fact that n ISl^2 2 (tr 9 8)^2 for any

symmetric 2-tensor Son an n-dimensional Riemannian manifold. D

To obtain an estimate for Q from the maximum principle, we need to
consider the ODE corresponding to (5.38), namely

d 2

dtq = q + rq.

The solution with initial data q(O) = q 0 < -r < 0 is

Crert

Get - 1'

where C ~ q 0 / (q 0 + r) > I. Therefore, applying the maximum principle to

equation (5.38) implies the following.

PROPOSITION 5.57. On a complete solution of the normalized Ricci flow

with bounded positive scalar curvature, there exists a constant C > 1 de-

pending only on go such that

a Crert

at logR- IVlogRl

2

= Q 2 - Cert_ 1.

This estimate for Q is known as a differential Harnack inequality.

Integrating it along paths in space-time yields a classical Harnack inequality:

a lower bound for the curvature at some point and time (x2, t2) in terms

of the curvature at some (x1, ti), where 0 :::; ti < t2. In particular, let

"(: [t 1 , t 2 ] --+ M^2 be a C^1 - path joining x1 and x2. Then by the fundamental

theorem of calculus,

R (x2, t2) 1t^2 d

log ( ) = -d (log R) b (t), t) dt

Rx1,t1 ti t

= l

1

t

2

( :t (log R) b (t), t) + (\!log R, ~;)) dt

rt2 ( Crert Id"( I)

2 lti IV log Rl

2

• Cert_ 1 - IV log RI · dt dt

> - - - - dt

1

t2 ( Crert 1 Id"( 1

2

)

• t 1 Cert - 1 4 dt

(5.39) = --11t2 Id __:!_^12 dt - log (Cert^2 - 1) +log (Gerti - 1).

4 t1 dt

Now define