Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 229

The coefficient of d/theta^2 is

[(

2

1 −tanh^2 (r/2)

(tanh(r/2))

)] 2

=

[

2sinh(r/2) cosh(r/2)

cosh^2 (r/2)−sinh^2 (r/2)

] 2

=sinh^2 (r/2)

while the coefficient of dr^2 is

2

1 −tanh^2 (r/2)

1

2cosh^2 (r/2)

=1,

so the metric reduces to

ds^2 D=dr^2 +sinh^2 (r)dθ^2.

The functionhmay then be written as

h(r)=

1

sinh(r) log(coth(r/2))

,

so that the metric ds^2 His

ds^2 H=h(r)

[

dr^2 + sinh(r)dθ^2

]

.

It will be convenient to have the formula for the curvatureκgof a metric
of the formg^2

[

dr^2 +(sinh^2 (r))dθ^2

]

.Itisgivenby

κg=−

[(

g′

g

)′

+1+

(

g′

g

)

coth(r)

]

.

Of course, it is in general difficult to decide what is really important in
all these formulas. The main point of the curvature formula is that it involves
two derivatives ing, and is close to−1 provided thatgis close to 1 and its
first and second derivatives are close to 0.
The main point abouthis that it is a function for which, asr→∞,we
have thath(r) is close to 1, whileh′andh′′are close to 0. This can be seen
by some simple uses of L’Hospital’s rule.
The idea of the proof of the theorem is now to find a functiong,whichis
equal to 1 for small values ofr, which is equal to h for large values ofr,and
for which the values ofκgstay close to−1. It is easy that one can do this,
provided thatg(r) agrees with 1 for large enough values ofrso thath(r)is
close to 1 and the first two derivatives ofhare close to 0. This then establishes
Theorem 6.1. The proof of Theorem 6.2 goes exactly the same way, reversing
the roles of ds^2 Hand ds^2 D.

12.8 The Spaghetti Model

We have seen how the Bollobas theory gives us a good picture of the spectral
behavior of a general Riemann surface. A reasonable question to ask is whether