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Contemporary Mathematics 432 CONTEMPORARY MATHEMATICS 432 In the Tradition of Ahlfors-Bers/ IV Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors http://dx.doi.org/10.1090/conm/432 In the Tradition of Ahlfors-Bers, IV CoNTEMPORARY MATHEMATICS 432 In the Tradition of Ahlfors-Bersl IV Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 14H15, 20H10, 28A75, 30F40, 30C62, 32G15, 54E40, 57M50. Library of Congress Cataloging-in-Publication Data Ahlfors-Bers Colloquium (4th : 2005 : Ann Arbor, Michigan) In the tradition of Ahlfors-Bers, IV : the Ahlfors-Bers Colloquium, May 19-22, 2005, Ann Arbor, Michigan / Dick Canary ... [et a!.], editors. p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 432) Includes bibliographical references. ISBN-13: 978-0-8218-4227-0 (alk. paper) ISBN-10: 0-8218-4227-7 (alk. paper) 1. Functions-Congresses. 2. Riemann surfaces-Congresses. 3. Mappings (Mathematics)- Congresses. I. Canary, Richard Douglas. II. Title. QA331.A462 2005 515--dc22 2007060747 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. org/ 10987654321 12 11 10 09 08 07 Contents Preface vii Plenary Talks IX Workshops and Workshop Speakers xi Colloquium Participants xiii Uniformly exponential growth and mapping class groups of surfaces J. W. ANDERSON, J. ARAMOYONA, and K. J. SCHACKLETON 1 An A 1 weight not comparable with any quasiconformal Jacobian CHRISTOPHER J. BISHOP 7 Curvature, stretchiness, and dynamics MOON DUCHIN 19 Some special loci in the Siegel space of genus two CLIFFORD J. EARLE 31 Another approach to the automorphism theorem for Teichmiiller spaces EGE FUJIKAWA 39 Energy of twisted harmonic maps of Riemann surfaces WILLIAM M. GOLDMAN and RICHARD A. WENTWORTH 45 Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis PETER HASTO, ZAIR IBRAGIMOV, DAVID MINDA, SAMINATHAN PONNUSAMY, and 8WADESH 8AHOO 63 From left earthquakes to right JuN Hu 75 Small eigenvalues and maximal laminations on complete surfaces of negative curvature CHRISTOPHER M. JUDGE 93 A generalized hyperbolic metric for plane domains LINDA KEEN and NIKOLA LAKIC 107 Subgroups of mapping class groups from the geometrical viewpoint RICHARD P. KENT IV and CHRISTOPHER J. LEININGER 119 Determinants of Laplacians, quasifuchsian spaces, and holomorphic extensions of Laplacians YOUNG-HEON KIM 143 v vi CONTENTS Hyperbolic and quasisymmetric structure of hyperspace LEONID V. KOVALEV and JEREMY T. TYSON 151 A classification of the modular transformations of infinite dimensional Teichmiiller spaces KATSUHIKO MATSUZAKI 167 Random hyperbolic surfaces and measured laminations MARYAM MIRZAKHANI 179 Extensions of holomorphic motions to quasiconformal motions SUDEB MITRA 199 Analyst's Traveling Salesman Theorems. A Survey RAANAN SHUL 209 Energy of harmonic maps and Gardiner's formula RICHARD A. WENTWORTH 221 Preface The Ahlfors-Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmiiller theory, hyperbolic manifolds, and partial differential equations. However, their work has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geomet- ric group theory, number theory and topology. These triannual colloquia serve as a venue to disseminate the work since the previous colloquium to the wider mathematical community and bring the key participants together to ponder future directions in the field. The third Ahlfors-Bers Colloquium was held in Ann Arbor, Michigan in May 2005 and was attended by nearly 150 mathematicians from eight countries. This was a particularly exciting time for the colloquium to take place as many old and well known problems had been resolved since the last colloquium. Here are some high- lights. In the theory of hyperbolic 3-manifolds, Ahlfors Measure Conjecture, the Bers-Sullivan-Thurston Density Conjecture, Marden's Tameness Conjecture and Thurston's Ending Lamination Conjecture had all been resolved, and new phenom- ena in the deformation theory of Kleinian groups had been revealed. The study of mapping class groups had been re-invigorated by techniques from geometric group theory and Teichmiiller theory. In the theory of meromorphic functions, the spher- ical Bloch constant had been determined. Versions of the uniformization theorem had been found that were purely metric in formulation, with applications to geo- metric group theory. The conformal invariance of percolation had been established. In addition to the plenary talks, there were three pre-colloquium workshops that focused on early career mathematicians. The workshops were Conformal Analysis and Geometric Function Theory organized by Lukas Geyer and Sergiy Merenkov, Deformation Theory of Hyperbolic 3-manifolds organized by Dick Canary and Jane Gilman, and Weil-Petersson Geometry of Teichmiiller space organized by Scott Wolpert and Mike Wolf. The list of plenary and workshop speakers and their titles is given below. The first Colloquium in this series was the Bers Colloquium (1995). This was followed by the first Ahlfors-Bers colloquium (1998) and the second Ahlfors- Bers colloquium (2001). Proceeding of these three colloquia have been published in the AMS Contemporary Mathematics Series, Lipa's Legacy, volume 211, In the Tradition of Ahlfors-Bers, volume 256, In the Tradition of Ahlfors-Bers, III, volume 335, respectively. One important part of the Ahlfors-Bers legacy is to cast a wide net that includes old and young, new and established researchers. We hope that with this volume the legacy of Ahlfors and Bers will continue to have an impact. vii viii 1. PREFACE AB05 was supported by the NSF and the University of Michigan. The orga- nizing committee consisted of Bill A bikoff, Dick Canary (Chair), Alex Eremenko, Jane Gilman, Juha Heinonen, Howie Masur, Yair Minsky, Michael Wolf, and Scott Wolpert. The editors of this volume and the organizing committee thank the mem- bers of the local organizing committee, Dick Canary, Lukas Geyer, Juha Heinonen, and Sergiy Merenkov, for making the colloquium a success. Dick Canary Jane Gilman J uha Heinonen Howard Masur Plenary Speakers with Titles Agol, Ian Tameness of negatively curved 3-manifolds Astala, Kari Complex analysis, impedance tomography and quasiconformal mappings Bishop, Chris Conformal mapping in linear time Bonk, Mario Conformal Analysis on Fractals Calegari, Danny Shrinkwrapping and the taming of hyperbolic 3-manifolds Kerckhoff, Steve Boundary value problems in deformation theory Minsky, Yair Ends, Laminations and Families of Surfaces Mirzakhani, Maryam Random hyperbolic surfaces and measured laminations Rees, Mary The Ending Laminations Theorem Direct from Teichmuller Geodesics. Souto, Juan Volume of hyperbolic manifolds and distances of Heegaard splittings Traizet, Martin Riemann surfaces with nodes and minimal surfaces without symmetries. ix Workshops and Speakers Conformal Analysis and Geometric Function Theory organizers: Lukas Geyer and Sergiy Merenkov Deformation Theory of Hyperbolic 3-manifolds organizers: Dick Canary and Jane Gilman Weil-Petersson Geometry of Teichmiiller space organizers: Scott Wolpert and Mike Wolfe Jeremy Best Huang, Zheng Constants for weighted The Canonical metric on a Riemann Moser-Trudinger inequalities on surface and its induced metric on the sub-Riemannian spaces moduli Brock, Jeffrey Ito, Kentaro Chains of flats, minimal irrational 3-dimensional extension of the Maskit laminations, and divergent geodesics in slice for once-punctured tori the Weil-Petersson metric Kim, Young-Heon Determinants of Laplacians and their Choi, Young holomorphic extensions Lines of minima in Teichmuller space Kovalev,
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