6 MARLIES GERBER

(2.3b) *. .(z) = - zp/2, p = p(i).

x 3 P

Let 3. . (z) = s, + is0. Note that

1,3 l

l

(2.4) (#T1.)*|Re z 2 dz| = |dsn |

1,3 ' ' !

and

2=1

(2.5)

(*T1.)*|Im

z

2

dz| = |ds,|.

Throughout this paper, we will let z = t + it2 =

peIT

denote smooth coordinate systems given by the charts

(p.,U.), i = 1,...,L, and we will let w = s, + is.=

rei9

denote the singular coordinate systems in stable sectors of

singular points given by the maps $. .o cp., i = l,...,m,

j = 0,...,p(i)- 1.

For r such that 0 r — ap , p = p ( i ), welet

D. . = {x€S? . : *. .(p.(x) = (s1 ,s ), where s2 + s2 r 2 }.

ijj

r

IJ IJ

i i L

J-

z

Let a1 be such that 0 af 1 and

(2.5a) [f(cp^12af) U r1(p:1Ba!)] c V ^ C * ^ ), i = l,...,m.

One further condition on af will be imposed in paragraph 2.7,

In each stable sector S? ., f is given in the (sn9s0)

coordinate system by

(2.6) (s1,s2) H-*(Xs1, i s ),

provided $~ . . ( s - , s

0

) € B , . Note t h a t (2.6 ) i s th e time-one

i j x

L

a