- CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 481
We define the function f : 1-l n V -+ IR by
(I.40) <]? (y) -:--/y (J (y)).
If we prove that f is a Lipschitz function, then (I.40) implies that S (p, r ( x))
near xis a Lipschitz hypersurface. (Let w = uyE1-lnV /y ((ay, by)), which isan open subset of V; then S (p, r (x)) n Wis a Lipschitz graph over 1-l n V
(using the vector field V).)To see that f is a Lipschitz function, we argue as follows. For any
y E 1-l n V, we have
Ir (ry (0)) - r (ry (J (y)))I = Ir (y) - r (x)I :S d (x, y)since /y (J (y)) E r-^1 (r (x)) and by the triangle inequality. From (I.39),
with r = /y, we obtain
Ir (ry (0)) - r (ry (J (y)))I 2': sine If (y)I,where E > 0 is as in (I.38). Hence using f (x) = 0, we have
. 1
(I.41) If (x)-f (y)I =If (y)I:::; sinEd(x,y),
which implies f is Lipschitz continuous at x.
Now we show the Lipschitz continuity of f at other points in 1-l n V. Let
V be the same smooth unit vector field in V as before. For any point Yl E
1-l n V, iet x1 =<I? (y1) ES (p, r (x)) n V. Consider the smooth hypersurface
1-lJ.-^1 c V defined by
1-£1 = {ry (J (y1)): y E 1-l n V such that f (y1) E (ay, by)}.
(The parts of the integral curves r to V between 1-lnV and 1-l1 have constant
length f (y 1 ):)^14 We have that 1-£ 1 passes through x1 and is transversal to
V. Repeating the above construction at x1 (instead of x), we have a map
<T?1 : 1-l1-+ S (p, r (x)) n V
(note that r (x1) = r (x)) and a Lipschitz function Ji : 1-£1-+ IR with
. <T?i(y) -=;= /y (Ji (y)).
Note that Ji (x 1 )·=·0. For the same reason that (I.41) is true, we have
. 1
(I.42) Iii (x1) - Ji (y)I :S -. -d (x1, y)
SlllE
for any y E 1-l1.
Given Yl as above, let
1-lo-=;= {y E 1-l n V: f (y1) E (ay, by)} c 1-l n V.
Note that the map
<I? 2 : 1-lo -:-+ 1-l 1,
defined by
<l?2 (y) = /y (J (y1)) ,
(^14) Let 1-l' ~ {y E 1-l n V: f (y1) E (ay, by)}. Then 1-£1 is a 'constant height' f (y1)
graph over 1-l' with respect to V.