1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 455

by (H.46). Applying Lemma H.37 to the concave function -f and by ap-

plying the uniqueness of V' f (I' 00 (so)) (i.e., Lemma H.36(iv)), we have

1 V' J (r oo (so)) v oo

JV 00 I = IV' f (r 00 (so)) I and IV' f (r 00 (so)) I - JV 001 •

Since we have proved that any sequential limit of the left-hand side of (H.45)
is the same as the right-hand side, the left limit exists and (H.45) holds. This
proves the lemma. D

4.4. Sharafutdinov's contraction map.

Let C C Mn be a compact connected locally convex set with nonempty

interior and boundary and let f E <!:Wo ( C), where <!:Wo ( C) is defined as in

(H.30). Let Cs~ {y EC: f (y) 2: s }, wheres E [O, !sup], as before. Let

I's : C-+ Cs

be defined by

(H.48) I's (x) ~ { ~x (s)

if f (x) 2: s,

ifj(x)<s,

where 'Yx [f (x), !sup] -+ C is the maximal integral curve for V' f /JV' f 1

2

emanating from x. Note that if x E Cs, then I's (x) = x, so by definition,

the map rs is a retraction..

The following is an analogue of Corollary H.10.

THEOREM H.59 (Sharafutdinov's distance nonincreasing retraction). As-

sume the notations above.

(1) (rs is length nonincreasing) For any s E [O, !sup] and any rectifiable

path a: [O, 1]-+ C, we have

L (a) 2: L (rs o a).

(2) (Continuity in x ands) The map

r: C x [O,fsup]-+ C,

defined by

r(x,s) ~rs(x),

is continuous.

PROOF. (1) From the definition of rectifiable path, it suffices to show

that for x, y E C and s E [a, b],

(H.49) de (rs ( x) , rs (y)) ~ de ( x, y).

Without loss of generality we may assume f (x) ~ f (y). There are three

cases to consider.

(i) Ifs~ f (x), then rs (x) = x and rs (y) = y, so that

de (I's (x) ,rs (y)) =de (x,y).