1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. APPROXIMATE BUSEMANN-FELLER THEOREM 483

(2) for any y E 1-l with d (x, y) = d (x, 1-l) and for any unit speed

minimal geodesic / : [O, d (x, y)] -+ M joining y to x there exists

E > 0 such that I ([O, c)) c n+.

We may define the set M-in the same way using n_ (so that M+ nM_ =

1-l). We leave it as an exercise to show that for any compact subset/(, of 1-l,

there exists 6 > 0 such that <I> (K x [O, 6)) c M+.

The second fundamental form II7-l of 1-l with respect to v is defined by

II7-l (X, Y) = (Dxv, Y) = - (v, Dx Y) for X, Y E T1-l.

Given E > 0, let Ni (1-l) denote the following subset of the open E-neighbor-

hood of 1-l on the side of v:

N/ (1-l) = {x EM+: d(x, 1-l) < c}.

Assuming the exercise in the previous paragraph, it is easy to see that, for

any E > 0 and for any compact subset K of 1-l, there exists 'fl > 0 such that

<I> (K x [O, 'fl)) c N 1 t (1-l).

Suppose that

(I.44)

for some constant 0 ~ A< oo, where 17-l is the first fundamental form of 1-l.

LEMMA I.42 (Nearest point projection map in small tubular neighbor-

hood). There exists a constant l(A, K,) > 0 with the following property. If

x E N;,tA,K-) (1-l), then there exists a unique point 7r (x) E 1-l closest to x, so

that the nearest point projection map

7r : N;,tA,K-) (1-l) -+ 1-{

is well defined. Moreover, 7r is smooth.

PROOF. Let

N + 1-l ~ {av ( x) : x E 1-l and a 2': 0}

denote the 'nonnegative normal bundle' of 1-l, where v (x) is the above choice
of unit normal at x. Consider the exponential map restricted to this bundle,
i.e.,

explN+7-l : N+1-l-+ M.

Then consider the parallel hypersurfaces

1-lr ~ {x EM+: d(x,1-l) =r},

where r 2': 0. Let IIr denote the second fundamental form of 1-lr, which is
well defined as long as 1-lr is smooth.
Given x E 1-l, let ax : [O, oo) -+ M be the unit speed geodesic with
ax ( 0) = v ( x). Recall the Ricca ti equation

\7 ax IIr = - (Rm (ax,·)·, ax) - n;