1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. BASIC SINGULARITY THEORY FOR RICCI FLOW 467

3.3. Trace Harnack inequality. Given a surface (M^2 , g) with posi-

tive curvature, the trace Harnack quantity is defined by

a

(Vl-5.35) Q = ~log R + R - r = at log R - IV' log Rl^2 •

(Also see Lemma 5.35 on p. 144 of Volume One.)

We have the following differential Harnack estimate of Li-Yau-Hamilton-

type. (See Corollary 5.56 on p. 145 of Volume One.)

COROLLARY A.42 (2d trace Harnack evolution and estimate). On any

solution of the normalized Ricci flow (A.19) on a complete surface with

bounded positive scalar curvature, Q satisfies the evolutionary inequality

(Vl-5.38)

a

at Q 2:: ~Q + 2 (V'Q, V' L) + Qz + rQ.

For the unnormalized flow (A.18), the analogous quantity

(Vl-p. 169a) Q -=~log R + R = at a log R - IV' log RI^2

satisfies

(Vl-5.57) :t Q 2:: ~Q + 2 ( V' Q' V' L) + {J2.

By the maximum principle,

(Vl-p.169b)

• 1
  Q (x, t) + t 2:: 0

for all x E M and t > 0.

In all dimensions, we have the following. (See Proposition 9.20 on p. 274
of Volume One.)

PROPOSITION A.43 (Trace Harnack estimate). If (Mn, g (t)) is a solu-

tion of the Ricci flow on a complete manifold with bounded positive curvature

operator, then for any vector field X on M and all times t > 0 such that

the solution exists, one has

(Vl-p.274)

aR R

8t + t + 2 (V'R,X) + 2Rc (X,X) 2:: 0.

The proof of Proposition A.43 will be given in Part IL When n = 2,
by choosing the minimizing vector field X = -R-^1 \i'R, it can be seen that
(Vl-p.274) is equivalent to (Vl-p.169b).
One also has Corollary 9.21 on p. 274 of Volume One, namely

COROLLARY A.44 (Trace Harnack consequence, tR monotonicity). If

(Mn,g (t)) is a solution of the Ricci flow on a complete manifold with

bounded curvature operator, then the function tR is pointwise nondecreasing

for all t 2:: 0 for which the solution exists. If (M, g (t)) is also ancient, then
R itself is pointwise nondecreasing.