Topology in Molecular Biology

224 R. Brooks

Proof.: Let us write the “pullback metric” of ds^22 by

f∗ds^22 =g^2 (z)ds^21 ,

whereg(z) is a real function, which will be zero at the critical points off.
The fact thatfis a holomorphic function is expressed here by the fact that
f∗(ds^22 ) is conformal to ds^21 .fwill be distance decreasing provided thatg<1.
The idea of the proof is as follows: letz 0 beapointatwhichgattains
its maximum value. Such a point has to exist, sinceS 1 is compact. Ifz 0 is
a branch point, theng(z)≡0, andfis a constant (clearly distance decreas-
ing). Otherwise, we may choose local coordinates aboutz 0 , and use the same
coordinates aboutf(z 0 ).
Writing in these coordinates

ds^21 =λ^21 (z)|dz|^2

and
ds^22 =λ 2 (z)|dz|^2 ,

we clearly have that

g(z)=

λ 2

λ 1

,

while

κ 1 =−

∆(log(λ 1 ))

λ^21

,

κ 2 =−

∆(log(λ 2 ))

λ^22

,

Atz 0 ,wehave

∆(log(g))≤ 0.

hence,

∆(log(g)) =∆(log(λ 2 ))−∆(log(λ 1 )) =−κ 2 λ^22 +κ 1 λ^21 ≤ 0

or

g^2 =

λ^22

λ^21

≤

(−κ 1 )

(−κ 2 )

< 1

Henceg<1, and the proof is complete.

Here is Alfors original version

Lemma 3.Letds^21 =ds^2 Handds 2 be two conformally equivalent metrics on
D,withκ 2 ≤−1=κ 1 .Thenanymapf:D→Dis distance nonincreasing
fromds 1 tods 2.