1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

376 7. THE REDUCED DISTANCE

9.5.1. Weak differentiability of the distance function d on a Riemannian

manifold. Let (.Mn, g) be a Riemannian manifold with sect (g) :::::: -K. Let

p EM and define dp (x) ~ d (p, x). Since IV'dp (x)I = 1 for a.e. x EM and

since the Hessian and Laplacian comparison theorems (A.9) hold pointwise

a.e., dp satisfies b..dp s (n - 1) ../K coth ( ../Kdp) in the support sense from

Definition 7.124(i). Applying Lemma 7.125 in the previous subsection to

dp (x), we obtain the following (see for example, [316], [246] or Theorem

1.128 in [111]).

LEMMA 7.126 (Laplacian comparison). Let (M,g) be a complete Rie-

mannian manifold with sect(§):::::: -K for some K:::::: 0.

(i) If K > 0, then b..dp::; (n - 1) ../K coth ( ../Kdp) in the weak sense.

(ii) If K = 0, then b..dp S nil p in the weak sense.

(iii) b..dp E Lfoc (.M).

(iv) f ;Vi dpb..cpdμg S f.,\/£ b..dp · cpdμg for any cp E c; (M).

REMARK 7.127. It is not clear to us whether b..dp E Lfoc for r > 1 when

dimension n :::::: 3. When n = 2, near the point p, we have b..dp f:. L[ 0 c for

r:::::: 2.

Below we give an example to show that in general,

for cp E c; (.M). Hence dp does not belong to the Sobolev space W^2 '^1 (.M).

Let Tn ~ [-1, 1 t / rv be the flat torus and choose p = 0. We consider

d; to facilitate our discussion. We have b..d; = 2n outside of the cut locus,
which is Cut (p) = 8 ([-1, lt)/,....,. For cp = 1 we have

r b..d;. cpdμ = 2n+ln -=!= 0 = r d;. b..cpdμ.

}7n }7n

The underlying reason for this phenomenon is the nonsmoothness of d; on

the cut locus of p. Let ~n ~ [-1+E,1 - E]n for E > 0 (which has piecewise

linear boundary). Let v denote the outward normal of {)~n C Tn. Then

r b..d;. cpdμ

Ir.n e

= f d;. b..cpdμ + f (V'd;. v) cpdμ - f d; C'v'cp. v) dμ.

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