Topology in Molecular Biology

214 R. Brooks

and it is easily seen that
∫∞

y 1

a^20

1

y^2

dy= (const)(y 1 )−^2 s.

Hence ∫∞

y 1

a^20

1

y^2

dy=

(

y 0

y 1

)∫∞

y 0

a^20

1

y^2

dy.

We would like to supply a similar analysis to the other terms. The treat-
ment of thean’s andbn’s is exactly the same, so we will focus on thean’s.
The idea is to apply the standard techniques of Sturm–Liouville comparison
to study the behavior of the solutions of the equations. We will do this in
several ways.
First of all, asygets large, the term−λ/ybecomes negligible in comparison
to the terms 4π^2 n^2. Thus, the differential equation foranhas two solutions,
one decaying like e−^2 πnyand the other blowing up like e^2 πny,forylarge.
In order to preserveL^2 -ness, the second term cannot appear. Thus, there is
a unique solution (up to constants)Fnof this equation, which decays like
e−^2 πnyforylarge. We may normalizeFnby insisting, say, thatFn(y 0 )=1.
We then compareFntoF 0 =y(1/2)−s. we find first of all thatFnis
positive for all values, and second thatFn/F 0 is a decreasing function ofy.
It follows that
∫∞
y=y 1 F

2

ny

− (^2) dy
∫∞
y=y 1 F
2
0 y−^2 dy

≤

∫∞

y=y 0 F

2

ny

− (^2) dy
∫∞
y=y 0 F
2
0 y−^2 dy

.

Rewriting this as

∫∞

y=y 1 F

2

ny

− (^2) dy
∫∞
y=y 0 F
n^2 y−^2 dy≤

∫∞

y=y 1 F

2

0 y

− (^2) dy
∫∞
y=y 0 F
2
0 y−^2 dy

.

and recalling that we have already evaluated the last term, gives

∫∞

y=y 1

a^2 ny−^2 dy≤

(

y 0

y 1

) 2 s∫∞

y=y 0

a^2 ny−^2 dy.

Summing over thean’s andbn’s now gives us the theorem.

12.4 Belyi Surfaces

We begin with the following

Definition 1.A compact Riemann surfaceSis called a Belyi surface if it
admits a holomorphic functionf :S→S^2 such thatf has at most three
critical values.