1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

J. SOLUTIONS TO SELECTED EXERCISES 501

which we also denote by b'Y. We have

b'Y ([(Bi, r)]) = 8 ~~ (s - dcone(S];) ([(B1, r)], "( (s)))

= s-+oo lim (s - Jr^2 + s^2 - 2(3rs)

= (3r,

which tends to -oo as r -+ oo since f3 < 0. We leave it to the reader to

show that, for any a E ( 1T, 21T) and t > 0 sufficiently small depending on a,

we have for the ray 'Y in (IR^2 , 90: (t)) that

r-+oo lim b'Y (B1, r) = -oo.

SOLUTION TO EXERCISE 1.38. Consider the hinge with geodesic sides

"!Vi and ai and angle ei at the common vertex Xi. By (I.33), we have

cos ei > cos ( 1T - c) = -cos c > -1.

Applying the hinge version of the Toponogov comparison theorem ( compar-

ing to Euclidean space and using the law of cosines), i.e., Theorem G.33(2),

we then have

. (J.6) r^2 (x) < df + rf + 2diri cosc.

Let a = sin c, so that cos c = Jl - a^2 and

(J.7) r^2 (x) < df + rf + 2diri~·

Now by (J.7), inequality (I.37) follows from establishing

a^2 c^2 ri ~ 2di ( Vl -a^2 c^2 - Vl -a^2 ).

Since ri ~^1 ~'° di, this is implied by

2 ( Jl - a^2 c^2 - Jf=li2)

a^2 c(l - c)

which in turn follows from

2: 1,

2(1+c) 2:22: J1-a2c2+J1-a2.

c

Inequality (I.37) has been proved.