1547671870-The_Ricci_Flow__Chow

14. MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 163

where K = R/2 is the Gaussian curvature and k = (T, 'VrN) is the geodesic
curvature of the curve 'Yp with unit tangent vector T and unit normal vector
N oriented outward to Mt.

PROOF. The derivatives with respect top are easily verified. By Lemma

3.11,

~L = -~1 Rds = -1 Kds.

at 2 '°Yp 'Yr

By the Gauss-Bonnet formula for a disc with outward unit normal N, one
has

27f = 27rX (M~) = { K dA =t= { (\lrT, N) ds

}Mt leMt

(5.51) = 1 ± K dA ± 1 (T, \lrN) ds,

Mp '°Yp

and hence

and

a

a A±= - f RdA= -47r± 21 kds.

t }M± p ,.., IP

LEMMA 5.91. Under the same hypothesis, L and A ± evolve by

a a^2

8tL= 8p2L

respectively.

Notice that these resemble heat equations.

PROOF. Differentiating (5.51) with respect to p yields

0 = 1 K ds + : 1 k ds.

'°Yp p '°Yp

Thus by Lemma 5.90,

a a 1 a

2

8tL = 8p "Ip kds = 8p2L.

The formulas for A± follow from the observation that

D

D