1547671870-The_Ricci_Flow__Chow

108 5. THE RICCI FLOW ON SURFACES

is an orthonormal frame field for g and

wl ~ eurt1 w2 ~ eu172

is an orthonormal coframe field for g. To compute the connection 1-form w~
for { ei} with respect to g, we begin by calculating

dw^1 = eu (d17^1 +du/\ 771 ) = eu (77^2 /\11i + h (u) 772 /\ 771 )

dw^2 = eu (d17^2 +du/\ 772 ) = eu (77^1 /\77I +Ji (u) 771 /\77^2 ).

Combining these equations with the general formula

(5.1) wi = dw^1 (e2, e 1) w^1 + dw^2 (e2, e1) w^2 ,

we obtain

wi = [77i (e1) + e-u h (u)] w^1 - [77i (e2) + e-u Ji (u)] w^2

= 11i + h (u) 771 - Ji (u) 772 ·

Then by applying this result and the second structure equation, we can write

the curvature 2-form of g as

Rm [gg = dwi = d11i + d [h (u) 771 - Ji (u) 772 ]

=Rm [hg + d [h (u)] /\ 771 - d [!1(u)]/\77^2 + h (u) d77^1 - Ji (u) d77^2

=Rm [hg + hh ( u) 772 /\ 771 - fif1 (u) 771 /\ 772

+ h (u) 17~ (!1) 772 /\ 771 - Ji (u) 77I (h) 771 /\77^2 ,

where we again used the identity d77k = 7JI (h) 771 /\ 772 implied by (5.1).

Because b.h u = \7 1 2 1, f 1 u + \7 1 2 2, f 2 u, this proves

Rm [gg =Rm [hg - (b.hu) 771 /\ 772

and hence

R 9 = 2K 9 = 2 Rm [gg (e1, e2) = 2e-^2 u (Kh - b.hu) = e-^2 u (Rh - 2b.hu).

D

To describe how the Laplacian changes, we begin with a general obser-

vation.

LEMMA 5.4. Let g (t) be a smooth I -parameter family of metrics on Mn.

Then

:tb.g(t) = - (:t9ij) \i'i\i'j - le (gij\i'i (%t9je)-~'Ve (gij :t9ij)) \i'k.

PROOF. If u is an arbitrary smooth function, we have

:t (b.u) = :t [gij (aiaj - rtak) u]

= (%tgij) \i'i'Vju-gij (%trfj) \i'ku+b. (%tu),