1547671870-The_Ricci_Flow__Chow

144 5. THE RICCI FLOW ON SURFACES

10.1. The case that R (-, 0) > O. Recall that the gradient Ricci soliton
equation ( 5. 7) implies that

(5.34)

Let L ~ log R and define

(5.35)

\7 R + R \7 f = 0.

Q ~ b..L+R-r.

Q is known as the differential Harnack quantity for a surface of posi-

tive curvature. On a gradient soliton of positive curvature, we can divide

equation (5.34) by R and take the divergence to find that Q = 0. As was

mentioned in Section 3, quantities which are constant in space on soliton so-

lutions often yield useful estimates on general solutions. For such solutions,

we shall obtain a lower bound for Q depending only on the initial metric go

by applying the maximum principle to its evolution equation.

LEMMA 5.53. Let (M^2 ,g(t)) be a solution of the normalized Ricci flow

on a compact surface with strictly positive scalar curvature. Then

a

(5.36) ot L = b..L + j\7 L J^2 + R - r.

PROOF. By (5.3), we have

:tL= ~:tR= ~(b..R+R(R-r))=b..L+J'\JLJ^2 +R-r.

D

COROLLARY 5.54. The differential Harnack quantity Q satisfies the iden-

tity

Q = :tL-J\7Ll2.

LEMMA 5.55. On any solution of the normalized Ricci flow on a compact

surface with strictly positive scalar curvature, one has

(5.37) a I^1

1

2

otQ = !::::.Q + 2 ('\i'Q, '\JL) + 2 \7'\JL + 2,(R-r)g + rQ.

PROOF. Using (5.5) with (5.36) and recalling the Bochner identity, we

calculate

gt Q = ( R - r) b..L + b.. ( :t L) + R ( :t L)

= b..Q + b.. J\7 Ll^2 + (R-r) b..L + R ( b..L + l\7 Ll^2 + R - r)

= !::::.Q + 2 (\'\lb..L, '\JL) + l\7'\i'Ll^2 )

+ 2R l\7 Ll^2 + (2R - r) !::::.L + R (R- r)

= b..Q + 2 ('\i'Q, \7 L) + 2 J\7\7 L l^2 + 2 (R - r) b..L + (R - r)^2 + rQ.

The result follows. D