1547671870-The_Ricci_Flow__Chow

152 5. THE RICCI FLOW ON SURFACES

length less than 7r/VKmax· Then since L(a1) = L(a2) < 7r/VKmax, there

are by Lemma 5.66 unique geodesic paths /3i and /32 between p and q that
are close to a1 and a2 respectively, hence close to ai and a2. Now as a
consequence of the Gauss Lemma (Lemma B.l), we have L (/3;) ~ L (ai)

for i = 1, 2. But then by Lemma 5.67, there is a smooth geodesic loop 'Y

with

L (!) ~ L (/3i) + L (/32) ~ L (a*) < L (a),

which is a contradiction. D

LEMMA 5.70. If 'Y is a weakly stable geodesic loop in an orientable Rie-

mannian surface (M^2 , g), then

1 Rds~O.

PROOF. Let T denote the unit tangent vector to 'Y· Since M^2 is ori-

entable, there is a well-defined unit normal N along 'Y· Since 'Y is geodesic

and n = 2, we have \lrN = 0, because

(\lrN, T) = T (N, T) = 0 and

1

(\lrN, N) = 2T (N, N) = 0.

Let 'Ye be a 1-parameter family of loops with 'Yo= 'Y and fhe/aB[e=o =

N. Then by the second variation formula (see 1.14 of [27])

:: 2 L(Te)I = 1 (R(N,T)N,T) ds = -1 R ds.

0=0 I I 2

Since 'Y is weakly stable, the result follows. D

Now let 'Yt be a smooth 1-parameter family of loops in a time-dependent

Riemannian manifold (Mn, g (t)). That is, each 'Yt is a smooth loop in Mn

depending smoothly on t in some open interval T. If Lt ( 'Y) denotes the

length of a loop 'Y with respect to the metric g (t), then by Lemma 3.11, we

have

(5.44)

d

d Lt('Yt) = ~ 1 ~g (T, T) ds -1 (\lrT, V) ds,

t It ut It

where Tis the unit tangent to 'Yt, V is the variation vector field 'Y* (a/at),

and ds is the element of arc length. This implies the following observation.

LEMMA 5.71. If (M^2 ,g(t)) is a solution of the normalized Ricci fiow

and 'Yt is a smooth 1-parameter family of geodesic loops, then

:t Lt bt) lt=T = ~ 1r (r - R) ds.

Combining this result with Lemma 5. 70 yields the following result.