1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

178 4. PROOF OF THE COMPACTNESS THEOREM

and that¢ (r) is nonnegative for r E [O, 1] when K::::; 0 or when K > 0 and

VK l'YI :S 7f. Assuming VK l'YI < 7f if K > 0, we compute

(IJI' ¢-IJI ¢')' = IJI" ¢-IJI ¢"

= ( ::2 IJI + K l'Yl

2

IJI) ¢ 2: 0

for all r E [O, l]. Integrating this from 0 tor gives us

IJI' (r) ¢ (r) - IJ (r)I ¢' (r) 2: IJI' (0) ¢ (0) + IJ (O)I ¢' (0) = 0,

that is, for r E (0, 1],

(4.15) IJI' (r) 2: IJ (r)I ~ (~;.

Hence

IJI !!:... IJI I 2: IJl^2 (1) ¢' (l)

dr r=l ¢ (1)

provided VK l'YI < 7f when K > 0. This proves

d I CSKl"l2 (1) 2

-g (\ly exp;^1 x, Y) = IJI -d IJI 2: / (l) !YI ·

r r=l snKli'l2

CSKI · 12 (1)

Note that snKli'l2^7 (1) is positive either when K :S 0 or when K > 0 and

VKl'Yol<1r/2. D

Recall that a 02 function¢ is (strictly) convex if its Hessian is positive

definite: \7V¢ > 0.

COROLLARY 4.46 (Local convexity of the distance squared function).

Suppose the sectional curvatures of (Mn,g) are bounded above by K. Then

the function f (y) ~ ~d^2 (x, y) is convex for any y E B ( x, 7f / ( 2VR)) not

in the cut locus of x.

PROOF. This follows directly from (4.14).

We also have

D

COROLLARY 4.47 (Convexity of small enough balls). Suppose the sec-

tional curvatures of (Mn, g) are bounded above by K. Then the ball B ( 0, r)

is convex if r :S min { inj 0, 7f / ( 2VR)}.

PROOF. Suppose x, y E B ( 0, r). Let / ( t) be the constant speed mini-
mal geodesic between x and y. We simply need to show that d (I (t), 0) < r
for every t. Consider the function f (z) = ~d (0, z)^2 • By Corollary 4.46, we
have that

Vt dtf d (1 (t)) = (\7^2 f) (d' dt' d1) dt > 0,