MARLIES GERBER

These two foliations have the same singularities x ,. . . ,x

and the same number ofprongs p = p(i) at each x., which

we will call a p-prong singularity of f. Moreover, there

e x i s t a 0 and a system of C°° coordinat e c h a r t s (p.,U.),

^ 1 1

i = 1,...,L, L m, on M such that U U- = M and for

i = 1, (cp.,U.) satisfies

/ (i) cpi(Ui)

(ii) cp.(x.) = 0

l l

(2.2) /

(iii) leaves of 7 get mapped to components of the

sets {Re zp = constant} f ) B

a

(iy) leaves of 7 get mapped to components of the

sets {Im 7T = constant} f l 2

a

u

(v) on U.,

(s

x and [i are given by

p-2 2-2

2 2

|Re z dz| and |Im z dz|, respectively

and for i m, (tp.,U.) satisfies

(i) (p.(U.) = (0,b.) x (0,c.) c (t ,t.)-plane,^21

i i i i

for some b.,c. 0

l l

(ii) leaves of 7 get mapped to segments

{tT = constant} f l cp.(U.)

1 ^i

I

(2 3 )\

(iii) leaves of J get mapped to segments

{t0 = constant} f l p.(U.)

2 ^i i

(iv) on U., \xS and iiU are given by |dt1|

V ^ and |dt2|9 respectively.

For convenience we assume that IL,...,U are pairwise disjoint.