Uniformization of Riemann surfaces : revisiting a hundred-year-old theorem PDF
Preview Uniformization of Riemann surfaces : revisiting a hundred-year-old theorem
Contents TheAuthors vii Foreword ix Generalintroduction xiii A Riemannsurfaces 1 I Antecedentworks 3 II Riemann 25 III RiemannsurfacesandRiemanniansurfaces 75 IV Schwarz’scontribution 97 Intermezzo 115 V TheKleinquartic 115 B Themethodofcontinuity 139 VI Fuchsiangroups 149 VII The“methodofcontinuity” 181 VIII Di↵erentialequationsanduniformization 199 IX Examplesandfurtherdevelopments 235 Intermezzo 283 X Uniformizationofsurfaces 283 C Towardsthegeneraluniformizationtheorem 321 XI Uniformizationoffunctions 333 XII Koebe’sproofoftheuniformizationtheorem 353 XIII Poincaré’sproofoftheuniformizationtheorem 361 Epilogue 377 Appendices 385 ThecorrespondencebetweenKleinandPoincaré 385 Somehistoricalreferencepoints 417 Bibliography 441 Index 471 The Authors AurélienAlvarez Universitéd’Orléans,UFRSciences,Bâtimentdemathématiques-RoutedeChartres,B.P.6759,45067Orléans Cedex2. ChristopheBavard InstitutdeMathématiquesdeBordeaux, UniversitéBordeaux1, 351coursdelaLibération, 33405Talence Cedex. FrançoisBéguin UniversitéParis13Nord,99avenueJean-BaptisteClément,93430Villetaneuse. NicolasBergeron InstitutdeMathématiquesdeJussieu, UMRCNRS7586, UniversitéPierreetMarieCurie, 4placeJussieu, 75252ParisCedex05. MaximeBourrigan DépartementdeMathématiquesetApplications,ÉcoleNormaleSupérieure,45rued’Ulm,75230ParisCedex05. BertrandDeroin DMAENSParisCNRSUMR8553,45rued’Ulm,75005Paris. SorinDumitrescu UniversitéNiceSophiaAntipolis,LaboratoireJ.-A.DieudonnéUMR7351CNRS,ParcValrose,06108Nice Cedex. CharlesFrances IRMAUniversitédeStrasbourg,7rueRenéDescartes,67084StrasbourgCedex. ÉtienneGhys UnitédeMathématiquesPuresetAppliquées,UMR5669CNRS-ENSdeLyon,46alléed’Italie,69007Lyon. AntoninGuilloux InstitutdeMathématiquesdeJussieu, UMRCNRS7586, UniversitéPierreetMarieCurie, 4placeJussieu, 75252ParisCedex05. FrankLoray InstitutdeRechercheMathématiquedeRennes,CampusdeBeaulieu,35042RennesCedex. PatrickPopescu-Pampu UniversitéLille1,UFRdeMaths.,BâtimentM2,CitéScientifique,59655,Villeneuved’AscqCedex. PierrePy IRMA,UniversitédeStrasbourg&CNRS,67084Strasbourg. BrunoSévennec UnitédeMathématiquesPuresetAppliquées,UMR5669CNRS-ENSdeLyon,46alléed’Italie,69007Lyon. Jean-ClaudeSikorav UnitédeMathématiquesPuresetAppliquées,UMR5669CNRS-ENSdeLyon,46alléed’Italie,69007Lyon. Foreword Mathematicianshaveambiguousrelationswiththehistoryoftheirdiscipline. They experience pride in describing how important new concepts emerged grad- ually or suddenly, but sometimes tend to prettify the history, carried away with imaginingsofhowideasmighthavedevelopedinharmoniousandcoherentfash- ion. Thistendencyhassometimesirritatedprofessionalhistoriansofscience,well awarethatthedevelopmenthasoftenbeenmuchmoretortuous. It is our implicit belief that the uniformization theorem is one of the major results of 19th century mathematics. In modern terminology its formulation is simple: EverysimplyconnectedRiemannsurfaceisisomorphictothecomplexplane, theopenunitdisc,ortheRiemannsphere. Andonecanevenfindproofsintherecentliteratureestablishingitbymeans of not very complicated argumentation in just a few pages (see e.g. [Hub2006]). Yetitrequiredawholecenturybeforeanyonemanagedtoformulatethetheorem and for a convincing proof to be given in 1907. The present book considers this maturationprocessfromseveralangles. Butwhyisthistheoreminteresting? Intheintroductiontohiscelebrated1900 article [Hil1900b] listing his 23 most significant open problems, David Hilbert proposed certain “criteria of quality” characterizing a good problem. The first of these requires that the problem be easy to state, and the uniformization theo- rem certainly satisfies this condition since its statement occupies only two lines! The second requirement — that the proof be beautiful — we leave to the reader to check. Finally, and perhaps most importantly, it should generate connections between di↵erent areas and lead to new developments. The reader will see how the uniformization theorem evolved in parallel with the emergence of modern algebraic geometry, the creation of complex analysis, the stirrings of functional analysis, the blossoming of the theory of linear di↵erential equations, and the birthoftopology. Itisoneoftheguidingprinciplesof19thcenturymathematics. And furthermore Hilbert’s twenty-second problem was directly concerned with uniformization. x Foreword Weshouldgivethereaderfairwarningthatthisbookrepresentsarathermod- est contribution. Its authors are not historians — many of them can’t even read German! Theyaremathematicianswishingtocastastealthyglanceatthepastof thissofundamentaltheoreminthehopeofbringingtolightsomeofthebeautiful —andpotentiallyuseful—ideaslyinghiddeninlong-forgottenpapers. Further- more,theauthorscannotclaimtobelongtothefirstrankofspecialistsinmodern aspects of the uniformization theorem. Thus the present work is not a complete treatise on the subject, and we are aware of the gaps we should have plugged if onlywehadhadthetime. Ourexpositionisperhapssomewhatunusual. Wedon’tsomuchdescribethe history of a result as re-examine the old proofs with the eyes of modern math- ematicians, querying their validity and attempting to complete them where they fail,firstasfaraspossiblewithinthecontextofthebackgroundknowledgeofthe periodinquestion,or,ifthatturnsouttobetoodi�cult,thenbymeansofmodern mathematicaltoolsnotavailableatthetime. Althoughtheproofswearriveatasa result are not necessarily more economical than modern ones, it seems to us that they are superior in terms of ease of comprehension. The reader should not be surprisedtofindmanyanachronismsinthetext—forinstancecallingonSobolev to rescue Riemann! Nor should he be surprised that results are often stated in a much weaker form than their modern-day versions — for example, we present thetheoremonisothermalcoordinates,establishedbyAhlforsandBersunderthe generalassumptionofmeasurability,onlyintheanalyticcasedealtwithbyGauss. Gauss’ idea seems to us so limpid as to be well worth presenting in his original context. We hope that this book will be of use to today’s mathematicians wishing to glancebackatthehistoryoftheirsubject. Butwealsobelievethatitcanbeused to provide masters-level students with an illuminating approach to concepts of greatimportanceincontemporaryresearch. The book was conceived as follows: In 2007 fifteen mathematicians fore- gatheredatacountryhouseinSaint-Germain-la-Forêt,Sologne,tospendaweek expounding to one another fifteen di↵erent episodes from the history of the uni- formization theorem, given its first complete proof in 1907. It was thus a week commemorating a mathematical centenary! Back home, the fifteen edited their individual contributions, which were then amalgamated. A second retreat in the same rural setting one year later was devoted to intensive collective rewriting, from which there emerged a single work in manuscript form. After multiple fur- ther rewriting sessions, this time in small subsets of the fifteen, the present book ultimatelymaterialized. WearegratefulforthefinancialsupportprovidedbythegrantANRSymplexe BLAN06-3-137237,whichmadetheabsorbingworkofproducingthisbookfea- sible.
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