Progress in Mathematics Volume 297 Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein (cid:88)(cid:105)(cid:97)(cid:110)(cid:122)(cid:104)(cid:101)(cid:32)(cid:68)(cid:97)(cid:105)(cid:32)• (cid:88)(cid:105)(cid:97)(cid:111)(cid:99)(cid:104)(cid:117)(cid:110)(cid:32)(cid:82)(cid:111)(cid:110)(cid:103) (cid:69)(cid:100)(cid:105)(cid:116)(cid:111)(cid:114)(cid:115) Metric and Differential Geometry The Jeff Cheeger Anniversary Volume (cid:69)(cid:100)(cid:105)(cid:116)(cid:111)(cid:114)(cid:115) (cid:88)(cid:105)(cid:97)(cid:110)(cid:122)(cid:104)(cid:101)(cid:32)(cid:68)(cid:97)(cid:105) Xiaochun Rong Department of Mathematics Department of Mathematics University of California Rutgers University Santa Barbara, New Jersey Piscataway, New Jersey USA USA ISBN 978-3-0348-0256-7 ISBN 978-3-0348-0257-4 (eBook) (cid:68)(cid:79)(cid:73)(cid:49)(cid:48)(cid:46)(cid:49)(cid:48)(cid:48)(cid:55)(cid:47)(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:48)(cid:51)(cid:52)(cid:56)(cid:45)(cid:48)(cid:50)(cid:53)(cid:55)(cid:45)(cid:52) Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012939848 © Springer Basel 2012 This work is subject to copyright. 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Printedonacid-freepaper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Dedication ............................................................... vii Preface .................................................................. ix Photos ................................................................... x Curriculum Vitae of Jeff Cheeger ......................................... xvii X. Dai and X. Rong The Mathematical Work of Jeff Cheeger, a Brief Summary .......... xix H.B. Lawson, Jr. The Early Mathematical Work of Jeff Cheeger ...................... xxxi Part I: Differential Geometry M.T. Anderson Boundary Value Problems for Metrics on 3-manifolds ............... 3 X. Chen and S. Sun Space of Ka¨hler Metrics (V) – Ka¨hler Quantization ................. 19 F. Reese Harvey and H. Blaine Lawson, Jr. Split Special LagrangianGeometry .................................. 43 Ch. Sormani How Riemannian Manifolds Converge ............................... 91 G. Tian Existence of Einstein Metrics on Fano Manifolds .................... 119 Part II: Metric Geometry P. Koskela and K. Wildrick Analytic Properties of Quasiconformal Mappings Between Metric Spaces .............................................. 163 A. Naor An Application of Metric Cotype to Quasisymmetric Embeddings ........................................................ 175 v vi Contents Part III: Index Theory J.-M. Bismut Index Theory and the Hypoelliptic Laplacian ....................... 181 X. Dai and R.B. Melrose Adiabatic Limit, Heat Kernel and Analytic Torsion ................. 233 X. Ma and W. Zhang Transversal Index and 𝐿2-index for Manifolds with Boundary ....... 299 W. Mu¨ller The Asymptotics of the Ray-Singer Analytic Torsion of Hyperbolic 3-manifolds ........................................... 317 J. Simons and D. Sullivan Differential Characters for 𝐾-theory ................................ 353 Dedicated to Jeff Cheeger on his 65th birthday Preface The articlesinthis volumeoriginatesfromlecturesgivenatthe internationalcon- ferenceonMetric and Differential Geometry, May11–15,2009.Heldatthe Chern Institute ofMathematics,Tianjinandthe Capital NormalUniversity,Beijing,the main goal of the conference is to bring together leading experts to survey the recent advances in metric and differential geometry. Major topics covered in the conference include metric spaces, Einstein manifolds, Ka¨hler geometry, geometric flows, index theory and hypoelliptic Laplacian, analytic torsions, and differential 𝐾-theory. This is reflected well in the collection in the volume. The conference wasalsousedasanoccasiontocelebratethe 65thbirthdayofJeffCheeger,whose immense influence can be felt in many of these fields. WeacknowledgethefinancialsupportofChernInstituteofMathematicsand Capital Normal University for the realization of the conference and NSF for the preparationofthese proceedings. We areverygrateful for all the refereesfor their careful job in reviewing the contributions to this volume and for their thoughtful suggestions for improvements. And we thank all the contributors to these proceedings. With submissions that they all could have sent for publication to excellent mathematical journals, they made possible the volume that we present here. Xianzhe Dai and Xiaochun Rong January 2012, Santa Barbara ix