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
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