1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

490 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

In Lemma I.47, parts (a), (b), and (c) correspond to parts (ii), (iii),

and (iv) of Lemma 1.5 of [107], respectively. In fact, a check of the proof

of Lemma 12.29 and the proof of Lemma 12.30, both in Part II, yields the

upper bounds in (a), (b), and ( c). As for the proof of the lower bound in

(a) and (b), it uses Lemma I.37 and Lemma I.34 regarding \l*r. We leave

the interested reader to either work out the proof as an exercise or see pp.
602-603 of [107].
One may ask how the gradients of the (smooth) mollified distance func-
tions are related to the (set-valued) generalized gradient of the (Lipschitz)
distance function. In this regard, we venture to guess the following first
approximation to a conjecture.

PROBLEM I.48 (Does !re: approach \l r in some sense?). As E-+ 0, does
the gradient \?re: approach the set gradient \l r in some sense? For example,
one may ask whether at any point x EM, !re: (x) limits to a vector in the
convex hull of (\l r) (x) in TxM. If so, is this limit uniform (on compact
subsets)? Note that away from the cut locus we have !re: -+ \Jr. Perhaps
another possibility is that !re: approaches a vector in the cone (with vertex
at 0) over the convex hull of (\l r) ( x) in the sphere s;;-^1.

Toward generalizing Proposition I.29 to large radii spheres (see Pre-
sumed Theorem I.50 below), one approach is to prove an approximate dis-
tance-nonincreasing result for the (smooth) level surfaces of re: and then take
the limit as E -+ 0 to obtain the distance-nonincreasing result for geodesic
spheres (i.e., level surfaces of r). One concern is how well the intrinsic dis-
tance function on a level surface of re: approximates the intrinsic distance
function ds(p,s) on a nearby level surface S (p, s) of r. The following example
illustrates a potential issue.

EXAMPLE l.49.

(1) (Bumpy hypersurfaces limiting to hyperplane) Consider the family

of smooth curves

in the plane, defined for E > 0. We have

lim Cc: =IR x {O}.

c:-70

The length along Cc: from (0,0) to (l,csin (c^2 )) is given by

Le:~ fo

1

\/1 + c-^2 cos^2 (c^2 x)dx

;::: 2-^1 fo

1

Jcos (2-^2 x) J dx.