1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

472 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

4. Critical point theory and properties of distance spheres

Continuing with the topic of the previous section, to study distance

spheres whose radii are not necessarily within the injectivity radius, we dis-

cuss some issues related to the nonsmoothness of the distance function r.

The main result of this section is that distance spheres with large enough

radii in complete noncompact manifolds with nonnegative sectional curva-

tures are Lipschitz hypersurfaces.

4.1. Critical point theory for the distance function.

In this subsection we briefly discuss critical point theory for the distance

function on a complete Riemannian manifold (Mn, g).

Fix p E M and let

r (x) ~ d (x,p).

The gradient of r, as a set (or set gradient), is defined.by

(V' r) (x) ~ {~ET M : [VI=^1 and for all u E [O, r (x)] we }.

• x have u + r ( expx ( -u V)) = r ( x) ·

By definition, \i'*r (p) = s;-^1 c TpM is the unit tangent sphere.

Observe the following properties of V' *r (p).

(1) If x is a smooth point of r, then

(\i'*r) (x) = (V'r) (x)

is the usual gradient.

( 2) A vector V E s;--^1 is in (V' * r) ( x) if and only if there is a unit

speed geodesic 'Y : [ 0, oo) ---+ M such that 'Y ( r ( x)) = x and

V=')'(r(x)).

(3) The set (V'*r) (x) C TxM is compact.

To see the only if part of (2), we note the following. Given V E

(\i'*r)(x), we have r(expx(-r(x)V)) = 0, so that expx(-r(x)V) = p.

Hence

d (expx (-r (x) V), x) = r (x),

so that

u f---t expx (-uV), u E [O, r (x)],

is a minimal geodesic joining x top. Property (3) follows from (2) easily.

Let

\l*r ~ LJ (V'*r) (x) C SM,

xEM

where SM ~ UxEM s;--^1 c TM denotes the unit tangent bundle.

LEMMA I.30 (Local compactness of V' * r). If {Vi} is a sequence in V' * r

and Voo E TM are such that Vi---+ V 00 , then

V 00 E V'*r.