Quasiconformal mappings and manifolds of negative curvature.pdf

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1、Quasicon~ormalmappingsandmanifoldsofnegativecurvaturePierrePansuCentredeMath~matiquesEcolePolytechniqueF-91128PalaiseauC~dex(U.A.169duCNRS)In1928~H.GrOtzsch[19]observedthattheclassicalLiouvilleandPicardtheorems-nonexistenceofentirefunctionswhichareboundedor~moregenerally,omitmorethantwopoints-e

2、xtendedtoaclassofnonholomorphicmaps.AholomorphicbijectionbetweenplanedomainsisconformalwithrespecttoEuclideanmetric:itsdiSferentialateachpointisasimilitude,i.e.anisometrytimesahomothety.H.Br~tzschconsideredmapswhosedifferentialsitsataboundeddistancefromsimilitudes.If~atsomepointxthedifferential

3、takesacircletoanellipsewithaxesaandb,hedefinedthedistorsionQ(x)astheleastnumbersuchthat1/Q~a/b!Q.Hefurthermoreallowedadiscretesetofsingularpointswherethemapisaramifiedcovering.HeshowedthatthePicardtheoremextendstotheclassofmapsdefinedonthewholeplanewhichhaveboundeddistorsion.Nowadays,thesemapsa

4、recalledq

5、isnotionplaysacrucialroleinG.D.Mostow~srigiditytheoremforcompactRiemannianmanifoldsofconstantsectionalcurvature-Ianddimension~3.Thistheoremstatesthattwosuchmanifolds,ifdiffeomorphic,havetobeisometric.Theargumentinvolvesquasiconformalmappingsinthefollowingway.AdiffeomorphismbetweentwoquotientsD/

6、PandD/r~oftheunitdiskDinRnliftstoaquasiconformalmapping~ofD.AextendsbycontinuitytoaquasiconformalhomeomorphismfoftheunitsphereS~-I.Itsatisfiesf-g:~(g)f,g~r(~)where~.denotestheisomorphisminducedbyonthefundamentalgroupsrandr~.Equation(~),togetherwithsomeregularityoffimpliesthatfisinfactconformal.

7、ThecorrespondinghyperbolicisometryofDdescendstoanisometrybetweenthequotients.Inthistalk,weinvestigatewhetherthefirststepsoftheargumentcarryovertogeneralsimplyconnectedmanifoldsofnegativecurvature.Inthisclass,thereisanaturalnotiono