8 MARLIES GERBER
The rn,r ,r are chosen so that 0 r r r 1;
#" .(D. . _ ) c p T (B
f
) , for i = 1., . . . ,m, j = 0, . . . ,p(i)- 1;
and r, is sufficiently small so that f and g agree for
9 2 2
s + s* near r . (Here we have dropped an unimportant constant
factor in (i),and we have simplified the choice of the
rvTs
i (30
a
f)
wwelett
e g0
m _i
from [G-K].) Outside U p . (3 3
f
), le g = f.
i = l
If p(i) 3 for all i Z {l,...,m}, we let g = gQ.
If not, then we need to do an additional "blow-up" procedure
(as in [Kl], §2 and [G-K], §8),which we now describe. Let
X be a function defined on [0,°°) such that
(i) x'(r) = r in a neighborhood of 0
(ii) X is C except at 0
(iii) X'(r) 0 for r 0
(iv) x'(r) = r for r r .
0
We define X : M -* M as fol3ows. For each i such that
p(i) = 1, let X be given in the polar coordinates
re19 =
Sl
+ is2 in U±(=
Ss±
Q) by
X(rei9) = X(r)eiG.
At all points x which are not in U(U.: p(i) = 1,i = l,...,m)9 iet
X(x) = x. Then let g = xSnx~ Condition (iv)on X ensures
that g = gQ = f at those points which do not correspond to
singular coordinates (s..,s9) (in any S. ., i = l,...,m,
j = 0,. . . ,p(i)-1) with s + s r .
The map g preserves a measure v given by
(2.9)
/ ds.,ds0 \ s
Q*4,* I
x
o o I in S. ., if p( i) 3 ,i = 1, . . . ,m, j = 0,. . .,p( i)-l;
x l , 3
v
P
i )
( 8
i
+
V
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