Table Of ContentMemoirs of the American Mathematical Society
Number 335
Charles L. Epstein
The spectral theory of
geometrically periodic
hyperbolic 3-manifolds
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
November 1985 • Volume 58 • Number 335 (first of four numbers)
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Table of Contents
Notation vi
1. Preliminaries 1
1.1 Introduction 1
1.2. Holomorphic Families of Operators 12
2. Floquet Theory 17
2.1 An Equivalent Family of Operators 17
2.2 A Second Equivalent Family of Operators 26
3. The Elliptic Case 32
3.1 The Analyticity of the Operators 32
3.2 An Estimate on the Derivatives of the Eigenvalues 35
3.3 A Lower Bound on the Density of the Absolutely
Continuous Spectrum 38
3.4 The Structure of the Spectrum Near Zero 48
4. The Parabolic Case 5 2
4.1 Introduction 5 2
4.2 The Analyticity of Lp 55
4.3 Boundary Behavior of L* 64
3
4.4 The Absolutely Continuous Spectrum of -A on H /T 82
4.5 The Asymptotic Behavior of ^-, (?) Near Zero 88
5. Applications of the Spectral Theory 102
5.1 Introduction 102
2 3
5.2 The L Theory of the Wave Equation in H /T 103
5.3 Lattice Point Asymptotics: The Exact Leading
Order Term 110
5.4 The Explicit Leading Term for £(R) 118
5.5 Asymptotics of the Lengths of Closed Geodesies 126
Appendices
1. Hyperbolic Manifolds and Hyperbolic Isometries 136
2. A Uniform Estimate for K (z) 148
v
3. Derivation of a Selberg Trace Formula 150
References 15 9
iii
ABSTRACT
In this paper we develop the spectral theory of the Laplace-Beltrami
3
operator for geometrically periodic hyperbolic 3-manifolds, H /G . Using
the theory of holomorphic families of operators, we obtain a quantitative
description of the absolutely continuous spectrum. The singularly continuous
spectrum is shown to be empty. The details of the structure of the lower
part of the spectrum are described. These results are obtained using the
Selberg trace formula and estimates for the eigenfunctions of elliptic
operators. They are applied to study the asymptotic density of an orbit
3 r i
of G in H , igp : g € Gi and also the asymptotics of the lengths of
3
certain classes of closed geodesies on H /G . The paper is a corrected
version of my dissertation submitted to New York University in June, 1983.
1980 Mathematics Subject Classification - Primary classification: 58G35;
secondary classifications: 35P10, 35P15, 35P20, 53C22, 53C35, 58G16, 10P45,
10D30.
Keywords - Spectral theory on manifolds, trace formulas, lattice point
asymptotics, geometrically non-finite groups.
Library of Congress Cataloging-in-Publication Data
Epstein, Charles L.
The spectral theory of geometrically periodic hyperbolic 3-manifolds.
(Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 335)
Corrected version of author's dissertation—New York University, 1983.
Bibliography: p.
"November 1985, volume 58 number 335 (first of four numbers)."
1. Three-manifolds (Topology) 2. Spectral theory (Mathematics) I. Title.
II. Series.
QA3.A57 no. 335 [QA613.2] 510s [514'.7] 85-21443
ISBN 0-8218-2336-1
IV
This memoir is lovingly dedicated
to my parents on the occasion of
their sixtieth birthdays.
ACKNOWLEDGEMENTS
I would like to thank my advisor, Peter Lax, for all he has taught me
and his encouragement and direction. I also would like to thank Peter
Sarnak and Percy Deift for their good counsels. To Troels J^rgensen, I owe
a great debt for inventing the beautiful geometrically periodic groups and
for the hours he spent explaining them to me.
I would like to express my gratitude to Ron and Giuliana Davidoff for
sustaining me through many hard months.
I would like to thank the three fine and patient people, Connie Engle,
Donna Fields and Gloria Lee, who expertly typed my thesis; and Madelyn
Verbeke for putting in many corrections. I also would like to thank
Dottie Phares for preparing the final manuscript, and the referee for his
careful reading and many important corrections.
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Notation
The Poincare Upper Half Space,
[ (x x , y) : y > 0 , (x x ) € 3R }
p 2 p 2
2 2 2 2 2
with the metric ds = (dx., + dx~ + dy )/y
The invariant hyperbolic volume element. In
(x,,x , y)-coordinates:
dx^dxdy
2
dVol =
3
y
A discrete subgroup of SL(2,C) .
3
A fundamental polyhedron for the action of T on H
The semidirect product of T with Z .
The semidirect product of T with TL . Every element g of
has a unique representation as gT where T € T generates
r*/r and g € r .
The conjugacy class of the element g . T may be omitted.
_*
The conjugacy class in 1 of an element g of l .
The Riemannian inner product on tangent spaces to hyperbolic
manifolds. In (x-,x,y) coordinates:
9
<X,Y> = (X Y +X Y +X Y )/y2 ,
1 1 2 2 3 3
where X = X,1 bx + X20Bux + X33 y and
l 2
1 x 2 x 3 y
l 2
vii
viii Charles L. Epstein
( , ) A Hilbert space inner product for the Hilbert space H . If H
H
2
is omitted, it is the inner product for L (X) X is a hyper
bolic manifold or an open set.
|| || The Hilbert space norm for the Hilbert space H ;
II f ll = <f,f)J/2 •
H
If H is omitted, then || || is the Hilbert space norm for
2
L (X) as above.
|| || The invariant length of a tangent vector to a hyperbolic manifold:
|| x || = <x,x>1/2 .
D(Q) or
I)(A) The domain of the quadratic form Q or the domain of the
operator A ; these are subsets of a Hilbert space.
grad u The invariant hyperbolic gradient of u . In (x..,x,y)
9
coordinates:
2 / bu ?\u bu \
grad u = y
[bx > bx > by )
l 2
cm
^ . The outward normal derivative of u . If Y, is the outward
K v
dv v
unit normal vector then:
|$= <Y ,gradu> .
v
Au The invariant hyperbolic Laplace-Beltrami operator. In
(x , x^, y)-coordinates:
1
A 2 / 5 u , d u S u \ Bu
Au = y \-^ + ri + 7i - y^ •
x?^x dx by ' J
1 2
In the following definitions X is either a hyperbolic manifold or an open
00
set in a hyperbolic manifold. C (X) is the space of compactly supported,
n
infinitely differentiable functions defined on X . If X Q Y is a
CO
closed set then C (X) is the space of infinitely differentiable functions
Spectral Theory of 3-Manifolds :
CO
defined on a neighborhood of X , if X is open then C (X) is the space
of infinitely differentiable functions defined on X .
2 °°
L (X) The closure of C (X) in the norm:
ft
||u|| = (J |u|2dVol)1/2 .
L X
W > (X) The closure of C°°(X) in the norm:
ull i o = (hH2 + || grad u||2]dVol)1/2 .
2 2 • —
W ' (X) The closure of C (X) in the norm:
II u || „ „ = (T [|u|2 + || grad u ||2 + |A |2]dVol)1/2 .
u
W > X
wk'2
loc The set of functions on X such that
S I (A-l)k/2(<j)u)|2dVol < co
X
for every (p ^ CQ(X) •
N-p(R;p^p) The lattice point counting function for T ;
1
#{g €r d(gp, ) < R} ;
: Pl
d(*,*) is distance measured w.r.t the hyperbolic metric.
Sometimes abbreviated to N(R) .
3 *
TT (x) The number of closed geodesies in a hyperbolic 3-manifold M /T
n
of length < x represented by elements in 1 of the form
Tn, (r* = r x^ X) .
g
