1547671870-The_Ricci_Flow__Chow

5. CONVERGENCE WHEN x (M^2 ) < 0

So for all t > 0 large enough, one gets

:t IV' Rl

2

:S 6. IY' Rl

2

+~IV R l

2

)

whence the result follows from the maximum principle.

121

D
Before treating the general case, we provide another example, in order
to illustrate the role played by the evolution of the Levi-Civita connection.

LEMMA 5.25. On any solution (M^2 , g (t)) of the normalized Ricci fiow,

IVY' R l^2 evolves by

:t IVY' Rl^2 = 6. IY'Y' R l^2 - 2 IY'Y'V' Rl^2 + (2R - 4r) IVY' Rl^2

• 2R (6.R)^2 + 2 ( V' R, V' IV' Rl^2 ).

PROOF. In dimension n = 2, the standard variational formula (3.3)

becomes

:trt = lgke [vi (:t911) + Y'1 (:t9ie)-ve (:t9ij)]

= -2 1(\i'iR<51. k + Y'1R<5i k - V' k Rgij ).

We find by commuting derivatives that

\i'iY'j6.R = 6. \i'iY'jR - 2R\i'iY'jR + ( R6.R + l 1v R l^2 ) 9ij - \i'iR\i'1R.

Thus we get

:t Y'iY'jR = Y'iY'j (:tR) - (:trt) \i'kR

= 6. \liY'jR + R6.Rgij + 2\i'iR\71R- r\i'iY'jR

and hence

at^8 IVVRI^2 =at o(·lJg ·ke viv )

1 RVkY'eR

= 6. IY'Y' Rl^2 - 2 IY'Y'V' Rl^2 + (2R - 4r) IVY' R l^2

+ 2R (6.R)^2 + 2 ( V' R, V' IV' Rl^2 ).

D

COROLLARY 5.26. If (M^2 , g (t)) is a solution of the normalized Ricci

fiow such that r < 0, then there exists C2 > 0 such that

IVY' Rl2 :S C2ert/2.

PROOF. By (5.19), there exists to ?: 0 such that R :S 0 for all t ?: to.

Applying (5.21) at such times, we find that there exists C~ > 0 such that

:t IVY' Rl^2 :S 6. IY'Y' Rl^2 - 4r IVY' Rl^2 + c~ ert/^2 1vv RI.