Progress in Mathematics 318 David Borthwick Spectral Theory of Infinite-Area Hyperbolic Surfaces Second Edition Progress in Mathematics Volume 318 Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Jiang-Hua Lu More information about this series at http://www.springer.com/series/4848 David Borthwick Spectral Theory of Infinite-Area Hyperbolic Surfaces Second Edition David Borthwick Department of Mathematics and Computer Science Emory University Atlanta, GA, USA ISSN 0743-1643 Progress in Mathematics ISBN 978-3-319-33875-0 DOI 10.1007/978-3-319-33877-4 ISSN 2296-505X (electronic) ISBN 978-3-319-33877-4 (eBook) Library of Congress Control Number: 2016939135 Mathematics Subject Classification (2010): 58J50, 35P25 © Springer International Publishing Switzerland 2007, 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This book is published under the trade name Birkhäuser The registered company is Springer International Publishing AG Switzerland (www.birkhauser-science.com) For Sarah, Julia, and Benjamin Preface to the Second Edition Producing a new edition has given me the chance to discuss some of the many interesting results that have been proven since 2007 New sections have been added to later chapters of the book describing these more recent advances in our understanding of resonance distribution and spectral asymptotics for hyperbolic surfaces In the last few years, we have also developed new techniques for the numerical computation of resonances A new final chapter has been added describing these methods The numerical computations are used to explore various conjectures related to resonance distribution While I have tried to incorporate as many new results as possible, the additions have been limited by the existing scope of the book For example, extensions of results that were already known for hyperbolic surfaces and more general manifolds in higher dimensions have not been included I have tried to update the notes at the end of each chapter to mention these developments (I apologize in advance for any errors or omissions in these notes.) One of the most promising new developments not covered is an alternative approach to meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds developed by Vasy [270, 271] This new method is particularly well suited to semiclassical (high-frequency) analysis and has already inspired some important new results A full expository treatment will appear in the forthcoming book of Dyatlov-Zworski [74, Ch 5] The new edition has also provided an opportunity to improve the organization in certain parts of the text The most prominent example of this is a change in context for the central part of the book, For the first edition, I limited the main text to exact hyperbolic quotients exclusively, in order to keep the presentation as simple as possible With the benefit of hindsight, it made sense to adopt the broader context of surfaces with hyperbolic ends for certain chapters, allowing the results in those sections to be stated in a stronger form I am extremely grateful to Catherine Crompton, Pascal Philipp, and Anke Pohl for providing lists of errata that needed to be fixed from the first edition I would also like to thank Pierre Albin, Kiril Datchev, Semoyn Dyatlov, Tanya Christiansen, vii viii Preface to the Second Edition Frédéric Faure, Colin Guillarmou, Peter Hislop, Dmitry Jakobson, Frédéric Naud, Peter Perry, Tobias Weich, and Maciej Zworski for helping me to keep abreast of the new developments in this area of research My thanks also go to Chris Tominich of Birkhäuser for encouraging the development of a second edition Atlanta, GA, USA February 2016 David Borthwick Preface to the First Edition I first encountered the spectral theory of hyperbolic surfaces as an undergraduate physics student, through the intriguing expository article of Balazs-Voros [13] on relations between the Selberg theory of automorphic forms and quantum chaos At the time, I was quite impressed at the range of topics represented, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and spectral theory In my previous experience, these were completely separate realms, but here they were all mixed together in the same setting Twenty years later, these topics not seem so far apart to me However, I am no less amazed by the rich cross-fertilization of ideas in this subject area The primary motivation for this book is the conviction that this sort of mathematics that bridges the divides between fields ought to be made accessible to as broad an audience as possible—to graduate students especially, for whom regular coursework often exaggerates the impression of boundaries between disciplines The spectral theory of compact and finite-area Riemann surfaces is a classical subject with a history going back to the pioneering work of Atle Selberg, who brought techniques from spectral theory and harmonic analysis into the study of automorphic forms These cases have been thoroughly covered in various expository sources In particular, Buser [51] develops the spectral theory for compact Riemann surfaces with a concrete approach based on hyperbolic geometry and cutting and pasting Most treatments of the finite-area case, for example, Venkov [272], emphasize arithmetic surfaces and connections to number theory For infinite-area hyperbolic surfaces, a good understanding of the spectral theory has emerged only recently The assumption of infinite area changes the character of the theory The resolvent of the Laplacian takes on a predominant role, and the emphasis shifts from discrete eigenvalues to scattering theory and resonances It has only been through dramatic advances in geometric scattering theory that the full development of the infinite-area theory has become possible My goal in this book is to present a relatively self-contained account of this recent development Although many of the results could be stated in greater generality (e.g., higher dimensions), the book is restricted to the hyperbolic surface context for ix 448 References 18 Beardon, A.F.: The exponent of convergence of Poincaré series Proc Lond Math Soc 18, 461–483 (1968) 19 Beardon, A.F.: Inequalities for certain Fuchsian groups Acta Math 127, 221–258 (1971) 20 Beardon, A.F.: The Geometry of Discrete Groups Springer, New York (1995) 21 Bedford, T., Keane, M., Series, C (eds.): Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces Oxford University Press, New York (1991) 22 Bérard, P.: Transplantaion et isospectralité I Math Ann 292, 547–560 (1992) 23 Bers, L.: A remark on Mumford’s compactness theorem Isr J Math 12, 400–407 (1972) 24 Bers, L.: An inequality for Riemann surfaces In: Chavel, I., Farkas, H.M (eds.) 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Quantum resonances and partial differential equations In: Proceedings of the International Congress of Mathematicians, vol III (Beijing, 2002), pp 243–252 Higher Education Press, Beijing (2002) Notation Guide WD ˛.T/ h; i ; / hzi B B.wI r/ BC zI r/ Cj C` C1 Dw dg d ; / ı dist ; / @0 F dh EX s/ F Fj F` G1 s/ Definition Asymptotic to (ratio approaches 1) Comparable to (ratio bounded above and below) Axis of the hyperbolic transformation T, see §2.1 Hilbert space inner product Distributional pairing S S ! C p C jzj2 for z C Hyperbolic unit disk, see §2.1 Ball in H of hyperbolic radius r, centered at w Ball in C of Euclidean radius r, centered at z Cusp component of X, see §6.1.1 Hyperbolic cylinder of diameter `, ` nH, see §2.4 Parabolic cylinder, nH, see §2.4 Dirichlet fundamental domain with center w H, see (2.13) Riemannian area form on a hyperbolic surface, see (2.8) Hyperbolic distance Exponent of convergence of , see (2.20) Euclidean distance in C or R Positive Laplacian on a hyperbolic surface Infinite ( D 0) boundary of compactified funnel, see §7.4 Measure induced on @X, see §7.4 Poisson kernel of X, see §7.4 Fundamental domain for , see §2.2 Funnel component of X, see §6.1.1 Funnel of diameter `, see §2.4 The entire function .s/G.s/2 , where G.s/ is Barnes G-function © Springer International Publishing Switzerland 2016 D Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, Progress in Mathematics 318, DOI 10.1007/978-3-319-33877-4 459 460 Notation Guide .z/ ` H Hs I.wI r/ `.T/ / LX L.s/ M.s/ MX m j A/ N0 nc nf NX r/ ord PX s/ ˚X s/ X t/ c f RX RX s/ SX s/ z; z0 / TX X s/ X t/ X s/ X X ZX s/ Gamma function Geometrically finite Fuchsian group, see §2.2 Cyclic hyperbolic group, see §2.4 Cyclic parabolic subgroup, see §2.4 Hyperbolic upper half plane, see §2.1 Hausdorff measure of dimension s, see (14.21) Shadow of B.wI r/ on @B, see (14.23) Displacement length of the hyperbolic transformation T, see §2.1 Limit set of , see §2.2 Primitive length spectrum of X, see §2.5 Parametric error term, see (6.11) Parametrix for s.1 s//, see (6.27) Moduli space of hyperbolic structures on X, see §2.7.2 Multiplicity of a resonance at , see (8.4) Patterson-Sullivan measure on /, see §14.1 j-th singular value of A, see §A.4 Nonnegative integers N [ f0g Number of cusps, see §6.1.1 Number of funnels, see §6.1.1 Resonance counting function, see (9.1) Multiplicity of a scattering pole at , see (8.24) Order of a meromorphic function at , positive for zeroes Hadamard product over the resonance set, see (9.71) Regularized trace of RX s/ RH s/, see (10.27) Counting function for length spectrum, see (2.18) Boundary-defining function for X, see §6.1.1 Cusp boundary-defining function, see §6.4 Funnel boundary-defining function, see §6.4 Resonance set, repeated according to multiplicity, see Ch Resolvent s.1 s// , see Ch Scattering matrix, see §7.4 cosh2 d.z; z0 /=2/ D Œ.x x0 /2 C y C y0 /2 =.4yy0 /, see §4.1 Teichmüller space of hyperbolic structures on X, see §2.7.2 Relative scattering determinant, see (9.67) Wave 0-trace, see (11.3) Regularized trace of RX s/ RX s/, see (10.3) Hyperbolic surface Radial compactification of X, see §6.1.1 Euler characteristic Selberg zeta function, see (2.23) Index Symbols 0-integral, 215 0-trace, 216 0-volume, 215 A analytic Fredholm theorem, 105, 427 asymptotically hyperbolic, 119 axis, 12 B Barnes G-function, 56 Bergman kernel, 378 Bers’ theorem, 308 boundary defining function, 103 Bowen-Series map, 372 C canonical curve system, 302 Carleman estimate, 126 circle, 10 collar, 306 compact core, 28, 100 compactification, 102 conformally compact, 104, 371 conservative, 330 convex, 16 convex cocompact, 29, 104 convex core, 25 critical line, cusp, 24, 28, 100 cuspidal, 30, 304 D Dirichlet domain, 16 sides, 20 displacement length, 12 distance, 10 duplication formula, 69 dynamical zeta function, 378 E Eisenstein series, 58, 132 elementary factor, 417 elementary hyperbolic surface, 19 elementary hyperbolic surfaces, elliptic transformation, 11 embedded eigenvalue, 124, 432 end, hyperbolic, 23 ergodic, 328 theorem, 331 escape rate, 365 essentially self-adjoint, 428 Euler characteristic, 7, 35 exponent of convergence, 32, 213 F Fan inequalities, 183, 435 Fenchel-Nielsen coordinates, 44 finite type, 187, 415 finitely generated, 20 finitely meromorphic, 105 fractal Weyl conjecture, 3, 408 Fredholm determinant, 439 © Springer International Publishing Switzerland 2016 D Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, Progress in Mathematics 318, DOI 10.1007/978-3-319-33877-4 461 462 Fuchsian group, 14 cocompact, 19 cofinite, 19 convex cocompact, 29, 104 elementary, 19 finitely generated, 20 geometrically finite, 19, 28 of the first kind, 19 of the second kind, 19 functional calculus, 67, 123, 431 fundamental domain, 15 funnel, 23, 28, 89, 100 resolvent, 90 funneled torus, 16, 404 G Gauss-Bonnet theorem, 34 generalized eigenfunctions, 57, 69, 131 geodesic, 10 geodesic normal coordinates, 12, 81 geodesic polar coordinates, 13 geometric limit, 304, 305 geometrically finite, 19, 28 Green’s function, 64 H Hadamard factorization, 419 Hadamard finite part, 214 Hausdorff dimension, 336 Hausdorff measure, 336 heat 0-trace, 312 heat invariants, 314 Hilbert-Schmidt, 438 Hopf-Birkhoff theorem, 331 horocycle, 24 Huber’s theorem, 53 hyperbolic distance, 10 ends, 23 surface, transformation, 11 unit disk, hyperbolic cylinder, 23, 81 resolvent, 83, 85 hyperbolic plane, resolvent, 65 resonances, 145 scattering matrix, 76 hyperboloid model, 37 I indicial equation, 124, 129, 135 Index isometric circle, 373 isospectral, 297 J Jensen’s formula, 188, 416 L Laplacian, 12 lattice-point counting function, 352 length counting function, 31, 347 length isospectral, 297 length spectrum, 31, 213, 298, 300 limit set, 15, 102 Lindelöf’s theorem, 187, 420 logarithmic residue theorem, 154 M Möbius transformations, Müller’s theorem, 61 Maass-Selberg relation, 217 marking, 43 McKean’s theorem, 54 min-max, 378, 435 minimum modulus theorem, 211, 243, 294, 420 moduli space, 44, 304 Mumford’s lemma, 54, 303 N Nielsen region, 25 truncated, 27 null-multiplicity, 154 O orbit equivalence, 372 order, 146, 415 ordinary points, 15, 102 P Pöschl-Teller potential, 85 pair of pants, 40 pants decomposition, 41 parabolic cylinder, 24, 94 resolvent, 95, 96 parabolic transformation, 11 parametrix, 107 Patterson-Sullivan measure, 321, 340, 385 pentagon rule, 40 Index Phragmén-Lindelöf theorem, 201, 205, 244, 422 Poincaré series, 32, 341 Poisson formula, 248, 258 Poisson kernel, 69 Poisson operator, 136 Poisson summation, 248, 424 prime geodesic theorem, 55, 347 prime number theorem, 347 primitive, 30 principal symbol, 444 properly discontinuous, 13 pseudodifferential operator, 443 classical, 444 R radial limit point, 324 resolvent, 2, 64, 99 kernel, 99 resonance, 2, 143 chains, 402 counting function, 3, 177, 185, 270, 281, 382 hyperbolic plane, 145 lower bound, 270 multiplicity, 145, 166 set, 3, 59, 143, 298, 300 upper bound, 177 Riemann sphere, Riemann surface, 15 right-angled hexagon, 37 Ruelle transfer operator, 376 S scattering determinant, 206, 216, 276, 284, 287 scattering matrix, 58, 76, 138 relative, 205 scattering phase, 277, 284, 394 scattering pole, 58, 150, 152 multiplicity, 152, 154, 166 scattering poles, 144 Schottky group, 14, 370 Schwartz function, 422 seam, 40 Selberg trace formula, 52, 59 Selberg zeta function, 33, 60, 213, 379 factorization formula, 213 functional equation, 57, 241, 246 self-adjoint, 428 shadow, 336 463 sine rule, 39 singular values, 182, 434, 445 smoothing operators, 443 spectral gap, 387, 411 spectral projector, 431 spectral theorem, 427 spectrum, 429 absolutely continuous, 67, 134, 433 continuous, 433 discrete, 123, 432 essential, 121, 432 point, 433 Stieljes integral, 273, 275, 350, 354, 357, 360 Stone’s formula, 431 stretched product, 116 surface, surface with hyperbolic ends, 100 surfaces with hyperbolic ends, symbol, 443 T Tauberian theorem, 348, 434 Teichmüller space, 302 Teichmüller space, 44 tempered distributions, 262, 423 three-funnel surface, 402 topological entropy, 365 topologically finite, 7, 20 trace-class, 437, 445 trapped set, 364 twist parameter, 43 U uniformization, 15 unique continuation, 124 W wave 0-trace, 249, 253, 258 Weierstrass factorization, 417 Weyl asymptotic law, 54, 60, 433 Weyl criterion, 432 Weyl’s inequality, 183, 436, 439 word length, 373 Wronskian, 86 X X-piece, 301 ... arithmetic surfaces and connections to number theory For infinite- area hyperbolic surfaces, a good understanding of the spectral theory has emerged only recently The assumption of infinite area changes... Borthwick, Spectral Theory of Infinite- Area Hyperbolic Surfaces, Progress in Mathematics 318, DOI 10.1007/978-3-319-33877-4_1 Introduction of the spectral theory (For geometrically infinite surfaces. .. finite -area surfaces, the theory was interpreted in terms of stationary scattering theory first by Faddeev [81] in 1967, and subsequently by Lax-Phillips [154] The spectral theory of hyperbolic surfaces