226 R. Brooks
Suppose that there exist constantsC 1 andC 2 such thatC 1 sup(κ 1 ) inf(κ 2 )≤sup(κ 2 )≤C 2 inf(κ 1 )≤C 2 sup(κ 1 )< 0.Then
C 2 ds^22 ≤ds^21 ≤C 1 ds^22Proof.After multiplyingds 1 by constants, we may apply the previous lemma.
It is clear that after choosing the constants appropriately, we may change the
role of ds 1 and ds 2. This completes the argument.
We like to paraphrase the corollary in the following way: “curvature close
and negative implies metric close.”
12.7 Large Cusps
In this section, we consider the following problem: letSObe a noncompact
finite-area Riemann surface, and letSCbe its conformal compactification. To
what extent are the hyperbolic metrics onSOandSCrelated?
It is not too diffcult to see that there need be no relationship in general.
For instance, ifSCis a sphere or a torus, andSOisSCwith several points
removed, thenSOwill carry a hyperbolic metric, whileSCwill not. Even
if bothSOandSCcarry hyperbolic metrics, they will look quite different –
SCis compact, whileSOis not. However, one gets the feeling that for many
geometric quantities arising in spectral geometry, the geometry does not really
see things that take place far away on small regions. In particular, it should
not make much difference to the body of the surface if a cusp is filled in or
not.
In this section, we see how to realize this feeling. The key notion is the
notion of large cusps:
Definition 2. 1. A cusp onSOis of length≥Lif there is a closed horocycle
about the cusp whose length is at leastL.
- The surfaceSOhas cusps of length≥Lif there is a collection of horocycles
with disjoint interiors, one enclosing each cusp, such that each horocycle
has length at leastL.
The main result of this section is then:
Theorem 6.For everyε, there existsL=L(ε)such that, ifSOis a finite-
area hyperbolic Riemann surface with cusps of length≥L, then outside stan-
dard cusp neighborhood onSOandSC, the hyperbolic metricsds^2 Oandds^2 C
satisfy
1
1+ε
ds^20 ≥ds^2 C≥(1 +ε)ds^20.