1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. ESTIMATES FOR CHANGING DISTANCES 43

where, as above, Z (t) is the compact set of all unit speed minimal geodesics

joining xo to x, with respect to the metric g (t), fort E [O, T]. The lemma

now follows from (18.1). 0

1.2. Heat operator acting on the distance function.

We are interested in the heat operator acting on the distance function.

In view of the fact that the distance function is only Lipschitz continuous,

recall the following.

DEFINITION 18.5 (Laplacian upper bound in the barrier sense). For a

function <p continuous in a neighborhood of a point x, we say that b.<p (x) :S

C in the barrier sense if for any c: > 0 there exists a C^2 function 'I/; defined

in a neighborhood of x such that 'I/; (x) = <p (x), 'I/; 2: <pin a neighborhood of

x, and b.'I/; (x) :SC+ c:.

We shall estimate b.gdg(xo, x) from above.

LEMMA 18.6 (Upper bound for b.d). Let (Mn,g) be a complete Rie-

mannian manifold. Given any xo E M and x E M - { xo}, we have

(18.4) b.gdg(xo,x) :S foso ((n-1) ((' (s))^2 -(^2 Rc ('y'(s),f''(s))) ds

for any unit speed minimal geodesic I': [O, so] --+ M joining xo to x and any

continuous piecewise C^00 function ( : [O, so] --+ [O, 1] satisfying ( (0) = 0 and

((so)= 1.

PROOF. First we shall construct an 'upper barrier' for the distance func-

tion dg ( ·, xo) in a neighborhood of x EM - {xo}. Let

c: =min { min (inj g (!' (s))), -

2

1

dg(xo, x)} > 0.

sE[O,so]

We extend I' to an n-parameter family of paths

bv }vEB(e:)

as follows (see below for notation), where B ( c:) is the ball of radius c: centered

at the origin in TxM and where 'Yo= 1· Given VE TxM, define

V (s) E T'Y(s)M, for s E [O, so],

to be the parallel translation of V = V (so) along {', with respect to the

metric g. We may define the continuous piecewise smooth vector field V

along I' by

(18.5) V (s) = ( (s) V (s) E T'Y(s)M,

where ( : [O, s 0 ] --+ [O, 1] is a continuous piecewise C^00 function satisfying
( (0) = 0 and ((so) = 1. Note that V (0) = 0 and V (so) = V. Then there
exists an n-parameter family of piecewise C^00 paths { f'v }vEB(e:) so that
(1) 'Yo (s) =I' (s) for s E [O, so],
(2) 'Yv (O) = xo,