1547671870-The_Ricci_Flow__Chow

160 5. THE RICCI FLOW ON SURFACES

SKETCH OF PROOF OF LEMMA 5.85. Let L and A denote length and

area, respectively, measured with respect to the metric g. Define 'r/ > 0 by

There exists co > 0 depending only on g and ri such that L ( 'Y) 2: co whenever
'Y is a smooth embedded loop such that CH("!):::; 47r-ri. By (5.47), we have

whence it follows that

for a = 1, 2. We also have

A (M^2 ) A (M^2 )

L (rv)I^2 < - 47r A(M2)^1.^2 - < 2KA (M^2 ).

This gives us some control on the curves 'Y· Now let ii be a sequence of

smooth embedded loops such that

By applying the curve shortening flow (in particular, Grayson's theorem [49]

that any embedded loop either converges to a geodesic loop or shrinks to a

round point) one may show that there exist a new sequence 'Yi of smooth

embedded loops such that

and

1

kJ ds:::; C,

'Yi

where ki denotes the Gaussian curvature of the curve 'Yi, and the constant

C < oo depends only on g and 'r/· From this, one shows that the 'Yi are

locally uniformly bounded in £^1 •^2 and C^1 •^112. One then concludes that for

any p < 2 and a < 1/2, there exists a subsequence that converges in L^1 ·P

and C^1 ·°' to an immersed curve 'Y=· Since 'Yoo minimizes length among all

nearby curves bounding regions of fixed areas, the formula for first variation

of arc length shows that 'Yoo has constant curvature. Hence 'Yoo is smooth.

We now show that 'Yoo is actually embedded. Since it is a smooth limit

of smooth embedded curves, it can at worst have points of self tangency.

If this is so, one can regard 'Y= as the union of two curves 'Yl and "( 2 of

positive lengths L1 and L2, respectively, bounding positive areas A 1 and

A2, respectively. Then the 'inside' of 'Yoo has area A 1 + A2. Let A3 be the