Topology in Molecular Biology

228 R. Brooks

wherefwill be a function that is equal to one outside of cusp neighborhoods.
We will want to choosefso that:

1. The metric ̃ds^2 Oextends to be a smooth metric across the cusps;


2. The curvatureκ( ̃ds^2 O) lies between the− 1 /1+εand−(1 +ε).
   The Alfors–Schwarz lemma, in the Wolpert version, will then guarantee

that the metrics ds^2 Candds ̃^2 Oare close. This will then establish the theorem.
To prove Theorem 6.2, we will proceed in the same way, reversing the roles
ofSOandSC. For this, we will need the noncompact version of the Ahlfors–
Schwarz theorem. We must choosefso that it gives us a complete metric near
the pointszi, in addition to the curvature estimates.
To set up the basic calculation, letDdenote the disk, and let ds^2 Dbe the
hyperbolic metric onD

ds^2 D=

4

(1−|z|^2 )^2

[dx^2 +dy^2 ].

We will also consider the hyperbolic metric ds^2 HonD−0. It is given by
the formula

ds^2 H=

(

1

−|z|log(|z|)

) 2

[dx^2 +dy^2 ].

Note that the ratio

h^2 =

ds^2 H

ds^2 D

is given by

h=

1

−|z|log(|z|)

2

1 −|z|^2

.

It will be convenient to take geometric coordinates. Letr(z) be the distance
from 0 in the metric ds^2 D. Then

r(z)=

∫|z|

0

2

1 −x^2

dx=

∫|z|

0

[

1

1+x

+

1

1 −x

]

dx= log

(

1+|z|

1 −|z|

)

,

which gives the inverse map as

|z|= tanh

(r

2

)

.

Writing

dx^2 +dy^2 =[d(|z|)^2 +|z|^2 dθ^2 ],

we have

4

(1−|z|^2 )^2

[dx^2 +dy^2 ]=

[

4

(1−tanh^2 (r/2))^2

]

×

[(

1

2cosh^2 (r/2)

) 2

dr^2 + tanh^2 (r/2)dθ^2

]

.