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