1547671870-The_Ricci_Flow__Chow

130 5. THE RICCI FLOW ON SURFACES

on a surface to compute the commutator [V'V', ~],obtaining

\i'i\i'j~f = \i'i\i'j\i'k\i'kf

= \7 i \7 k \7 J \i'k f - V' i ( Rje 'Ve f)

= V'k\i'i'VJV'kf-RfkJ\i'e\i'kf-RieY'JY'ef

• RJeY'i\i'ef - \i'iRJe\i'ef

= ~\i'i\i'Jf-V'k (RfkJY'd)-RfkJ\i'e\i'kf

• ~e\i'JY'ef - RjeY'i\i'ef - Y'iRJe\i'ef
  1
  = ~ Y'i Y'Jf - 2 (V'iR\7 Jf + V'if'VJR - (\7 R, \7 f) 9iJ)


• 2R ( V'iV'Jf - ~ (~!) 9iJ) ·

Combining these results, we get

a 1

at Mij = ~ \i'i\i'Jf - 2 (~R) 9iJ + (r - 2R) MiJ

= ~ ( \7 i \7 J f - ~ ( R - r) 9iJ) + ( r - 2R) MiJ.

D

COROLLARY 5.35. On a solution (M^2 , g (t)) of the normalized Ricci

flow, the norm squared of the tensor M evolves by

(5.22) ~ IMl^2 = ~ IMl^2 - 2 IV' Ml^2 - 2R IMl^2.

at

PROOF. Recalling Lemma 3.1 and using the result above, we obtain

a 2 a ( ·k ·e )

at IMI = at l gJ MiJMke

= 2 (M, ~M + (r - 2R) M) + 2 (R - r) IMl^2

= ~ IMl2 - 2 IV'Ml2 -2R IMl2.

D

Equation (5.22) is the key result that motivates the following strategy.

If we can prove that R ~ c for some constant c > 0 independent oft, we

will obtain an estimate of the sort

Then we can consider the modified Ricci flow

(5.23)

a

atg = 2M = 2\7\7 f - (R - r) g = (r - R) g + LV'f9·