1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

474 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

EXAMPLE I.32 (Spheres and tori). (i) The sphere Sn (p) of radius p.
Given p E sn (p), the set of regular points of r is sn (p) - {p, -p} and the
only critical points are p and -p.
(ii) For 2-dimensional rectangular tori we have that for each point p the
distance function r has exactly 3 critical points (Exercise).^6

In view of (I.25), we give the following definitions. Given 0 E (0, ~],

define

Ce (x) ~{VE s~-l: ,{ (V, W) < 0 for all WE (V'*r) (x)}.

Since Ce ( x) is the intersection of convex subsets of s;;-^1 (balls of radii ~ ~),

it is convex inside s;;-^1. Equivalently, the cone on Ce (x) with vertex at the

origin is convex inside TxM·

We say that x is a 0-regular point of r if Ce (x) is nonempty. In

particular, x is a regular point of r if and only if xis a 7r /2-regular point of

r. Note that, since (V'*r) (x) is compact, if xis a 0-regular point of r, then

for c: > 0 sufficiently small, x is also a ( 0 - c: )-regular point of r.

Lemma I.30 also yields the following.

LEMMA I.33 (Regular points form an open set). For any 0 E (0, ~], the

set of 0-regular points of r is an open set. Moreover,

Ce~ LJ Ce (x) c SM

xEM

is an open set.

PROOF. We prove that SM - Ce is a closed set. Let Vi E SM - Ce

be a convergent sequence with Vi -t V 00 ; then the corresponding basepoints

converge Xi -t x 00 • By the definition of Ce, there exist Wi E (V'*r) (xi) such

that

L (Vi, Wi) 2: 0.

Passing to a subsequence, by Lemma I.30 we may assume

Wi -t Woo E (V'*r)(x).

Since L (Voo, Woo) 2: 0, we have V 00 E s;;-^1 - Ce (x 00 ).

4.2. Geometric characterization of regular points.

The following is the main result of this subsection.

D

LEMMA I.34 (Characterization of regular points, I). A point x E M-{p}

is a regular point of r if and only if there exist V E s;;-^1 , c > 0, and c: > 0

such that

r (expx (sV)) - r (x) 2: cs

for alls E [O, c:), i.e., there exists a direction in which the distance is growing

at least at a positive rate.^7

(^6) See also Examples 1.6-1.8 on pp. 3-4 of [29].
(^7) We may think of such a Vas a gradient-like vector.