Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 227

Fig. 12.11.A simple example

We remark that part of the statement of the theorem is that, in the pres-
ence of large cusps,SChas a hyperbolic metric. One can see from Gauss–
Bonnet that this will be the case ifL> 2 π. In effect, filling in a cusp takes
away 2πfrom the Gauss–Bonnet integrand. A horocycle of lengthLbinds
a region of areaL. So if all the horocycles have length≥L> 2 π, there is
enough area left over so thatSChas a negative Euler characteristic, and hence
a hyperbolic metric.
This argument is reasonably sharp. Let (G,O) be the graph given in
Fig. 12.11. Then, as we have seen in Sect. 12.5,S^0 (G,O) is the equilateral
torus with one puncture, and the standard horocycle onSO(G,O) has length
6, which is just a little bit less than 2π.ButSC(G,O) is a torus, and hence
doesn’t carry a hyperbolic metric.
Theorem 6.1 has a converse:

Theorem 7.For everyε, there existsR=R(ε)with the following property:
LetSCbe a compact hyperbolic surface, andz 1 ,...,zkpoints onSCsuch
that the injectivity radii about thezi’s are at leastR, and such that the balls
B(zi,R)are disjoint.
LetO=SC−{z 1 ,...,zk}, with its hyperbolic metric.
ThenSohas cusps of length≥sinh (1/(1 +ε)R), and outside of cusp
neighborhoods, we have

1

1+ε

ds^20 ≥ds^2 C≥(1 +ε)ds^20.

The strategy of the proof of Theorem 6.1 is as follows: we will consider two
conformally equivalent metrics onSC. The first metric will be the hyperbolic
metricds^2 ConSC. The second metric will be of the form

d ̃s^2

O=f

(^2) (z)ds 2
O,