Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 225

The proof will be a combination of the two arguments above. We again look
at the functiong(z), and at an interior maximumz 0 it is clear thatg(z 0 )≤1,
by the same curvature calculation as before. But how do we produce an interior
maximum?
The idea is to look at the family of functions

fr(z)=f(rz)

forr<1, and the corresponding functionsgr. We may take the disk{z:|z|<
r}as coordinates to see thatgrgoes to 0 at the boundary. This is because in
this coordinate, clearlyλ 1 (z)→∞as|z|→r, whileλ 2 remains finite.
Hence, ifgr(z) is not identically 0, it must have an interior maximum, so
gr(z)≤1 everywhere. But clearlygr(z)→g(z)asr→1, so we conclude that
g(z)≤1 everywhere. This gives the lemma.
Here is the version we will be using.

Theorem 5.Letf:S 1 →S 2 be a holomorphic map between two (not neces-
sarily compact) Riemann surfacesS 1 andS 2 ,andletds 1 andds 2 be metrics
onS 1 andS 2 respectively.
Suppose that the metricds 1 is complete, and that

sup

z∈S 2

κ 2 (z)<inf

z∈S 1

κ 1 (z)≤sup

z∈S 1

κ 2 (z)< 0.

Thenfis distance nonincreasing.

Proof.After passing to the universal coverings, we may assume thatS 1 and
S 2 are bothD.
We now writefr(z)=f(rz) as before. The argument is exactly the same
as in Ahlfors’ argument, except at one small point. As before, the function
grmust go to zero at the boundary, from completeness of ds 1. The only
difference is that instead of using the pointwise estimateκ 2 ≤−1, we have
to make do with comparisons of curvature at different points. Thus we must
replace pointwise curvature estimates with sup and inf estimates.
This concludes the proof.

We note at this point that the mapf no longer plays much of a role.
There is no loss in assuming that instead of two Riemann surfaces and a
map between them, we deal with one Riemann surface and two conformally
equivalent metrics on it. The role of the function can be replaced by allowing
the second metric to degenerate at some points. Here is our final version of the
Ahlfors–Schwarz lemma. We present it as a corollary to the previous version.
We would like to think of this as the “geometer’s version” of Ahlfors–Schwarz,
because it gives a nice, clean geometric statement, but from the analyst’s point
of view it may miss a lot that is covered by the lemma.

Corollary 2.LetSbe a Riemann surface with two complete metricsds 1 and
ds 2 , which are conformally equivalent.