
teach mueller space

ganita saastramlo, a (real) topological (ledhaa differential) uparitala suutram _ 2 yokka techmullar space formula _ 1, anedi formula _ 2 paie sanklishta nirmaanaalanu isotopic ayina homomorphisms charya varaku paramiti chese sthalam. teachmueller pradaesaalaku oswald teachmueller paeru pettaru.
oka teachmueller antariksha suutram _ 1 looni prati binduvunu "gurtinchabadina" reeman uparitalaala yokka isomorphism taragatigaa pariganinchavacchu, ikkada oka "marking" anedi suutram _ 2 nundi daaniki chendina homomorphisms yokka isotopy taragati. uparitalampai gurtinchabadina hyperbolic nirmaanam kosam dheenini madyuli spacega chudavachu, mariyu idhi sahaja topologyni istundi, dheeni kosam idhi genus formula _ 7 yokka uparitalam kosam dimension formula _ 6 yokka bantiki homomorphicga untundi. eee vidhamgaa teachmueller spacenu reeman madyuli space yokka universal cover arbifoldga chudavachu.
teach mueller sthalam praamaanika sanklishtamaina aneka retlu nirmaanam mariyu sahaja kolamanala sampadanu kaligi undhi. eee vividha nirmaanaala yokka rekhaganita lakshanaala adhyayanam anedi kriyaaseela parisoedhana.
techmueller sthalaanni adhyayanam chese ganita shaastra upa-rangaanni techmueller siddhaantam antaaru.
charitra.
genus formula 9 yokka uparitalampai sanklishta nirmaanaala vaividhyaalanu vivarinchadaaniki formula _ 6 paaraamitulu avasaramani telisina bernhard reeman (1826-1866) chesina pani nundi reeman uparitalaalu mariyu sambandhita fusian samuuhaala kosam madyuli pradaesaalu adhyayanam cheyabaddaayi. pantommidava sataabdam chivaralo-iravayyava sataabdam praarambhamlo, teak mueller sthalam yokka prarambha adhyayanam rekhaaganitamgaa undhi mariyu reeman uparitalaalanu hyperbolic uparitalaalugaa vyakhyaaninchadam meeda sthaapinchabadindi. pradhana sahakaarulalo felix clain, henry paayinkaare, paal kobe, jacob nielsen, robbert fricay mariyu verner fenchel unnaaru.
madyuli adhyayanaaniki teachmueller yokka pradhana sahakaram eee amsaaniki quasicone formal mappinglanu pravesapettadam. munupati, marinta praathamika rachanalalo laeni adanapu lakshanaalanu andinchadam dwara madyuli pradaesaala adhyayanaaniki marinta lothugaa ivvadaaniki avi anumatistaayi. rendava prapancha yuddham taruvaata eee amsaanni eee vislaeshanhaatmaka siralo marinta abhivruddhi chesaaru, mukhyamgaa lars ahlforce mariyu lipman bers. eee siddhaantam churukuga konasagutondi, techmueller space (bers pravesapettina) yokka sanklishta nirmaanam gurinchi aneka adhyayanalu jarigaayi.
1970la chivaralo vilium thurston chesina krushi taruvaata, tech mueller space adhyayanamlo rekhaganita sira punaruddharinchabadindi, athanu oka uparitalam yokka mapping class group yokka adhyayanamlo upayoginchina rekhaganita sankshepananu pravesapettaadu. eee samuuhaaniki sambandhinchina itara mishrama vastuvulu (mukhyamgaa vakra samudaayam) kooda tech mueller sthalaaniki sambandhinchinavi, mariyu idhi rekhaganita samuuha siddhaantamlo chaalaa churukaina parisoedhana amsam.
nirvachanalu.
sanklishta nirmaanaala nundi techmullar space.
suutram _ 2 oka orientable mruduvaina uparitalam (dimension 2 yokka differentiable manifold) gaa undanivvandi. anadhikarikamga formula _ 2 yokka teach mueller space formula _ 1 anedi formula _ 2 nundi isotopy varaku reeman uparitala nirmaanaala sthalam.
adhikarikamga dheenini eee krindi vidhamgaa nirvachinchavacchu. difiomorphism formula _ 16 unte formula _ 2paie rendu sanklishta nirmaanaala formula _ 14 samanamga untundih
appudu suutram _ 1 anedi eee sambandham kosam suutram _ 2 paie sanklishta nirmaanaala samanatva taragatula sthalam.
maroka samaanamaina nirvachanam eee krindi vidhamgaa undih suutram _ 1 anedi jatala suutram _ 25 yokka sthalam, ikkada suutram _ 17 anedi reeman uparitalam mariyu suutram _ 27 anedi difiomorphism, mariyu suutram _ 29 anedi holomorphic difiomarphysaniki isotopic ayithe rendu jatala suutram _ 28 samanamga pariganinchabadutundi. atuvanti jatanu "marked reeman surface" anie pilustaaru; "marking" anedi difiomorphism; markingla yokka maroka nirvachanam vakrarekhala vyavasthala dwara.
uniformisation siddhaantam nundi ventane lekkinchabade rendu saadhaarana udaaharanalu unnaih golam suutram _ 30 (reeman golam chudandi) paie oka pratyaekamaina sanklishta nirmaanam undhi mariyu suutram _ 31 (sanklishta vimanam mariyu unit disk) paie rendu unnaayi mariyu prati sandarbhamlo saanukuula defiomorphisms samuuham kudinchadaginadi. anduvalla suutram _ 30 yokka techmullar space oke point mariyu suutram _ 31 sarigga rendu paayintlanu kaligi untundi.
konchem ekkuva prameyam unna udaaharana open annulus, dheeni kosam teach mueller space anedi virama suutram _ 34 (suutram _ 35 kii sambandhinchina sanklishta nirmaanam reeman uparitala suutram _ 36).
torus mariyu flat kolamanala yokka teachmueller sthalam.
tadupari udaaharana torus suutram _ 37, eee sandarbhamlo edainaa sanklishta nirmaanaanni sanklishta sankhya suutram _ 39 kosam suutram _ 38 (sanklishta deerghavruttaakaara vakrata) yokka reeman uparitalam dwara grahinchavacchu, ikkada
idhi sanklishtamaina eguva sagam-plain. appudu manaku oka bijection untundih
anduvalla suutram _ 43 yokka tetch mueller space suutram _ 44
euclidean plaentoe suutram _ 45nii gurtinchinatlayite, techmullar pradaesamlooni prati binduvunu suutram _ 46paie gurtinchabadina chadunaina nirmaanamgaa kooda chudavachu, anduvalla techmullar sthalam jatala suutram _ 47thoo bijectionlo untundi, ikkada suutram _ 48 anedi chadunaina uparitalam mariyu suutram _ 49 anedi suutram _ 50paie isotopy varaku difiomorphism.
parimita rakam uparitalaalu.
ivi techmullar space ekkuvagaa adhyayanam cheyabade uparitalaalu, veetilo moosivesina uparitalaalu untaayi. oka uparitalam oka compact uparitalaaniki diphiomarphic mainus oka parimita samitiki unte adhi parimita rakaniki chendinadi. suutram _ 2 anedi jaathi suutram _ 52 yokka moosivesina uparitalam ayithe, suutram _ 2 nundi suutram _ 53 paayintlanu tolaginchadam dwara pondina uparitalam saadhaaranamgaa suutram _ 55 gaa suuchimchabadutumdi mariyu suutram _ 56 dwara daani tetch mueller spacenu suuchistumdi.
teach mueller khaaliilu mariyu hyperbolic kolamanalu.
paina perkonnavi kakunda prati parimita rakam orientable uparitalam sthiramaina vakrata suutram _ 57 yokka puurti remanian kolamanalanu angeekaristundi. parimita rakam yokka icchina uparitalaaniki uniformisation siddhaantam nundi eee krindi vidhamgaa kolamanalu mariyu sanklishta nirmaanaala madhya bijection untundi. anduvalla formula _ 58 ayithe, techmullar space formula _ 59 nuu formula _ 53 kasplatho genus formula _ 52 yokka gurtinchabadina hyperbolic uparitalaala samitigaa grahinchavacchu, idhi jatala formula _ 62 samiti, ikkada formula _ 48 anedi hyperbolic uparitalam mariyu formula _ 64 anedi difiomorphism, maadyulo suutram _ 67 isometriki isotopic ayithe formula _ 62 mariyu formula _ 66 gurtinchabade samanatva sambandham.
techmullar sthalampai topology.
paina lekkinchina anni sandarbhaallo, teach mueller sthalampai spashtamaina sthalaakruthi undhi. saadhaarana sandarbhamlo suutram _ 1 nuu sthalavignaanam cheyadaaniki aneka sahaja margalu unnaayi, bahusha athi saralamainadi hyperbolic kolamanalu mariyu podavu vidhula dwara.
suutram _ 69 suutram _ 2 meeda closed karv mariyu suutram _ 71 gurtinchabadina hyperbolic uparitalamaite, oka suutram _ 72 suutram _ 48 (paramiti varaku) paie pratyaekamaina closed geodesic suutram _ 73 kii homotopic avutundi. (homotopy taragati) suutram _ 69kii anubandhinchabadina "podavu function" yokka suutram _ 75 vadda viluva appudu
suutram _ 78 suutram _ 2 paie saadhaarana moosivesina vakrarekhala samitigaa undanivvandi. appudu mapp
idhi oka embedding. space formula _ 82 utpatti topologyni kaligi undhi mariyu formula _ 1 prerepita topologyni kaligi undhi. eee topology suutramtoe _ 84 suutram _ 85 kii homomorphic
vaastavaaniki suutram _ 86 vakratalato oka embeddingnu pondavacchu, mariyu suutram _ 87 kooda pondavacchu. rendu sandarbhaalloonuu paina perkonna homomorphism yokka rekhaganita rujuvunu ivvadaaniki embeddingnu upayoginchavacchu.
chinna teach mueller pradaesaalaku marinni udaaharanalu.
moodu-randhraala golampai parimita parimaanam yokka pratyaekamaina puurti hyperbolic metrik undhi mariyu anduvalla sthira vakrata suutram _ 88 yokka parimita-parimaana puurti kolamanala yokka tech mueller space oka point (idhi munupati paeraa yokka parimaana suutram nundi kooda anusaristundi).
techmullar space formula _ 89 mariyu formula _ 90 sahajamgaa eguva sagam-plainga grahinchabadataayi, fenchel-nielsen coordinatelanu upayoginchi chudavachu.
techmueller space mariyu conformal structures.
hyperbolic kolamanala sanklishta nirmaanaalaku badulugaa conformal nirmaanaalanu upayoginchi teach mueller sthalaanni nirvachinchavacchu. nijaaniki, conformal structures rendu (vaastava) kolatalalo sanklishta nirmaanaalatoo samanamga untaayi. antekakunda, ekaruupata siddhaantam uparitalampai prati conformal class remanian kolamanalalo sthiramaina vakrata yokka pratyaekamaina metrik undani kooda suuchistumdi.
techmullar praatinidhya pradaesaalugaa khaaliilu.
techmullar space yokka maroka vivarana uparitala samuuhaalaku praatinidhya sthalamgaa undhi. suutram _ 2 hyperbolic ayithe, parimita rakam mariyu suutram _ 92 suutram _ 2 yokka praathamika samuuham ayithe, appudu teak mueller space sahaja bijectionlo untundih 
mapp gurtinchabadina hyperbolic structure formulanu _ 62nii kuurpu formula _ 101kii pamputundi, ikkada formula _ 102 anedi hyperbolic structure yokka monodromy mariyu formula _ 103 anedi formula _ 50 dwara prerepinchabadina isomorphism.
idhi suutram _ 1 nuu suutram _ 106 yokka closed upasamitigaa grahistundani gamaninchandi, idhi dheeniki topologyni istundi. homeomorphism suutram _ 107 nii neerugaa chuudataaniki dheenini upayoginchavacchu.
tichmullar sthalam yokka eee vivarana adhika tichmullar siddhaantam dwara saadhaaraneekarinchabadindi, ikkada samuuha suutram _ 95 ekapaksha ardha saralamaina abaddham samuuham dwara bharti cheyabadutundi.
vargaalapai oka vyakhya.
paina perkonna anni nirvachanaalanu differentiable manifolds vargaaniki badulugaa topological vibhaagamlo cheyavachu, mariyu idhi vastuvulanu maarchadu.
ananta-dimensional teachmueller khaaliilu.
parimita rakam kaani uparitalaalu kooda hyperbolic nirmaanaalanu angeekaristaayi, veetini ananta-dimensional spacela dwara paramiti cheyavachu (homeomarphic nundi formula _ 109). tichmullar siddhaantaaniki sambandhinchina ananta-dimensional space yokka maroka udaaharana uparitalaala dwara lamination yokka tichmullar space.
mapping class group yokka charya mariyu madyuli space kuu sambandham.
madyuli space kuu mapp.
techmullar space nundi madyuli space varaku reeman uparitalaala madyuli space varaku formula _ 2 kii diphiomarphic mapp undhi, idhi formula _ 111 dwara nirvachinchabadindi. idhi cover mapp, mariyu formula _ 1 kevalam anusandhaaninchabadinanduna idhi madyuli space kosam arbifold universal cover.
mapping taragati samuuham yokka charya.
suutram _ 2 yokka mapping taragati samuuham anedi suutram _ 2 yokka difiomorphism samuuham yokka coaset group formula _ 114, idhi gurtimpuku isotopic ayina vaati yokka saadhaarana upa samuuham dwara (adhe nirvachanam difiomorphismlaku badulugaa homeomarphismlatho cheyavachu mariyu uparitalaala kosam, idhi phalita samuuhaanni maarchadu). defiomorphisms samuuham sahajamgaa tichmullar space meeda panichestundi 
suutram _ 117 oka mapping taragati mariyu suutram _ 118 daanini soochinche rendu defiomorphisms ayithe avi isotopic. anduvalla tetch mueller pradaesamloe formula _ 119 mariyu formula _ 120 taragatulu okate, mariyu paie charya mapping class group dwara karakamga untundi.
teach mueller sthalamlo mapping class group formula _ 114 yokka charya sarigga discontinuesga untundi, mariyu coashant madyuli space.
sthiramaina paayintlu.
mapping class group yokka edainaa parimita upasamuuham techmular pradaesamloe global fixed point (anni group elements dwara fixes cheyabadina point) undaa anie nielsen saakshatkaara samasya adugutundi. marinta shaastreeya padaalaloo prasna emitante formula _ 114 yokka prati parimita upasamuuhaanni formula _ 2 (ledhaa konni sanklishta nirmaanam yokka holomorphic defiomorphisms samuuhamgaa samanamga) paie konni puurti hyperbolic metrik yokka isometryla samuuhamgaa grahinchavacchaa? dheenini steven kerkhoff parishkarinchaaru.
samanvayalu.
fenchel-nielsen coordinatelu.
techmullar space formula _ 1 looni fenchel-nielsen coordinatelu (verner fenchel mariyu jacob nielsen paeru pettabadinavi) uparitala formula _ 2 yokka pyantu kullipovataniki sambandhinchinavi. idhi formula _ 2 nuu jatala pantlaga kullipovadam, mariyu kullina prati vakrarekhaku daani podavu hyperbolic metriklo techmueller space looni paayintku anugunamgaa untundi, mariyu twist anie piluvabade maroka nijamaina paramiti nirvachinchadaaniki marinta nimagnamai untundi.
genus formula _ 52 yokka closed surface vishayamlo pyantu kullinappudu formula _ 128 vakratalu untaayi mariyu manaku formula _ 6 paaraamitulu labhistaayi, idhi formula _ 130 yokka parimaanam. fenchel-nielsen coordinatelu vaastavaaniki homomorphism formula _ 131nii nirvachistaayi.
puncture unna uparitalam vishayamlo konni jatala pyaantlu "degenerate" avtayi (vatiki cusp untundi) mariyu rendu podavu mariyu twist paaraamitulanu matrame istayi. mallii eee sandarbhamlo fenchel-nielsen coordinatelu homomorphism suutraanni _ 132 gaa nirvachistaayi.
kotha coordinatelu.
suutram 133 ayithe uparitala suutram 134 aadarsha tribhujaalanu angeekaristundi (dheeni sheershaalu sarigga puncturelu). euler lakshanaaniki suutram prakaram atuvanti tribhujaaniki suutram _ 135 tribhujaalu untaayi. suutram _ 2 paie unna hyperbolic structure formula _ 48 anedi prati tribhujaanni hyperbolic ideal tribhujaaniki pampe (isotopy varaku pratyekamainadi) difiomorphism formulanu nirnayistundi, tadwara suutram _ 1 loe oka binduvu. atuvanti nirmaanaaniki paaraamitulu tribhujamlo antukunna tribhujaala yokka prati jatha bhujaala anuvaada podavulu. formula _ 140 atuvanti paaraamitulu unnaayi, ivi prati okkati formula _ 141 loe edainaa viluvanu teesukogalavu, mariyu nirmaanam yokka paripuurnata oka sarala sameekaranaaniki anugunamgaa untundi mariyu tadwara manaku saraina parimaana suutram _ 142 labhistundi. eee coordinatelanu "kotha coordinatelu" antaaru.
muusina uparitalaala kosam, oka jatha pantlanu rendu aadarsha tribhujaala kalayikaga kullinchavacchu (dheenini moodu-randhraala golampai asampoorna hyperbolic metrikga chudavachu). eee vidhamgaa manaku suutram _ 130 paie suutram _ 128 kotha akshaamsaalu kooda labhistaayi.
bhookampaalu.
techmullar pradaesamloe oka saadhaarana "bhookampa maargam" anedi oke kotha ledhaa podavaina fenchel-nielsen coordinatenu (uparitalam yokka sthira aadarsha tribhujaaniki) marchadam dwara nirnayinche maargam. aadarsha tribhujaalu ledhaa pantlanu tectonic plaetlugaa mariyu kothanu plate motionga chudatam dwara eee paeru vachindi.
marinta saadhaaranamgaa geodesic laminationla venta bhookampaalu cheyavachu. thurston siddhaantam appudu teach mueller pradaesamloe rendu binduvulu oka pratyaekamaina bhookampa maargamtho kalustaayani perkondi.
vislaeshanhaatmaka siddhaantam.
quasicone formal mappings.
rendu reeman uparitalaala madhya oka quasiconformal mapping anedi oka homomorphism, idhi uparitalampai sarihaddu paddhatiloo conformal nirmaanaanni vaikalyam chestundi. marinta khachitamgaa idhi daadaapu pratichota bhedinchadaginadi mariyu "vistarana" anie piluvabade sthiramaina suutram _ 145 undhi, antey
ikkada suutram _ 147 anedi conformal coordinate suutram _ 148 mariyu daani samyoga suutram _ 149 looni utpannaalu.
prati isotopy taragatilo pakshika-anadhikaarika mappinglu unnaayi, kabatti teach mueller sthalaaniki pratyaamnaaya nirvachanam eee krindi vidhamgaa undhi. suutram _ 2kii reeman uparitala suutram _ 17 difiomarphicnu saricheyandi, mariyu tech mueller sthalam gurtinchabadina uparitala suutram _ 152thoo sahaja bijectionlo untundi, ikkada suutram _ 153 anedi quassicon formal mapping, paina perkonna adhe samanatva sambandham varaku untundi.
quadratic differentials mariyu bers embedding.
paie nirvachanamtho, suutram _ 154 ayithe, techmullar sthalam nundi suutram _ 97-equivarient parishkarala sthalam varaku beltramy avakalana sameekaranaaniki sahaja patam untundi. ivi schwarzian utpannam dwara suutram _ 17 paie chaturasraakaara bhedaalaku daariteestaayi. vaati sthalam sanklishta parimaana suutram _ 128 yokka sanklishta sthalam, mariyu teach mueller sthalam yokka chitram oka open sett. eee pataanni bers embedding antaaru.
suutram _ 17 paie oka chaturasraakaara bhedaanni suutram _ 17kii anugunamgaa unna anuvaada uparitala dwara suuchimchavachchu.
teachmueller mappings.
rendu gurtinchabadina reeman uparitala suutram _ 25 mariyu suutram _ 161 madhya suutram _ 163 yokka isotopy taragatilo ellappuduu oka pratyaekamaina quasiconformal mapping suutram _ 162 untundani, idhi kaneesa vistarananu kaligi untundani teach mueller siddhaantam perkondi. eee pataanni teach mueller mapping antaaru.
rekhaganita chitramlo dheeni ardham prati rendu diphiomarphic reeman uparitalaalaku suutram _ 14 mariyu difiomorphism suutram _ 165kii suutram _ 14nii soochinche rendu bahubhujalu unnaayi mariyu okadaanikokati pampe affine mapp undhi, idhi anni quasicopharmal maps suutram _ 162loe atichinna vistarananu kaligi untundi.
kolamanalu.
teach mueller metrik.
suutram _ 168 mariyu vaati madhya unna teachmueller mapping vistarana suutram _ 169 kaligi unte, vaati madhya unna teachmueller dhooram nirvachanam suutram _ 170 dwara untundi. idhi vaastavaaniki suutram _ 1 paie dooraanni nirvachistundi, idhi daani topologyni prerepistundi mariyu daani kosam adhi puurtavutumdi. idhi saadhaaranamgaa techmueller space yokka metrik jyaamiti adhyayanam kosam upayoginche metrik. mukhyamgaa idhi rekhaganita samuuha siddhaantakartalaku aasakti kaligistundi.
adevidhamgaa nirvachinchabadina oka function undhi, quassicon formal dilatationlaku badulugaa hyperbolic uparitalaala madhya myapla yokka lipsitz sthiraamkaalanu upayoginchi, suutram _ 172 paie, idhi samaruupata kaadu.
veyil-petterson metrik.
reeman uparitala suutram _ 17 paie chaturasraakaara bhedaalu suutram _ 174 nundi teach mueller sthalam vadda sparsha binduvutoe gurtinchabadataayi. veyil-petterson metrik anedi quadratic differentialspy 175 antargata utpatti suutram dwara nirvachinchabadina remanian metrik.
compact.
adhyayanam cheyabadina teach mueller pradaesaala yokka aneka asamaanamaina sankshepanalu unnaayi. munupati sankshepanalalo chaalaa varaku techmueller pradaesamloe oka binduvu empikapai aadharapadi untaayi, kabatti modular samuuham krinda avi asthiramgaa undavu, ivi asoukaryamgaa undavacchu. vilium thurston taruvaata eee pratikuulata lekunda sankshepananu kanugonnadu, idhi atyanta vistrutamgaa upayoginche sankshepanagaa maarindi.
thurston compactification.
techmullar pradaesamloe prati binduvuku saadhaarana closed karvla yokka hyperbolic podavulanu chudatam dwara mariyu (ananta-dimensional) projective pradaesamloe moosivetanu teesukovadam dwara, anantam vadda unna paayintlu projective kolichina laminationlaku anugunamgaa unde compactification nuu pravesapettaaru. sanklishtamaina sthalam closed bantiki homomorphicga untundi. eee thurston sankshepana modular samuuham dwara nirantaram panichestundi. mukhyamgaa modular samuuhamlooni edainaa moolakam thurston yokka sankshepanalo oka sthira binduvunu kaligi untundi, dheenini thurston modular samuuhamlooni muulakaala vargeekaranalo upayoginchaadu.
bers compactification.
bers compactification anedi techmullar space yokka embedding bers yokka chitraanni moosiveyadam dwara ivvabadutundi, dheenini adhyayanam chesaaru. bers embedding anedi teach mueller pradaesamloe oka binduvu empikapai aadharapadi untundi, kabatti modular samuuham krinda idhi asthiramgaa undadu, vaastavaaniki modular samuuham bers compactificationpy nirantaram panicheyadu.
teachmueller compactification.
teachmueller sankshepanalo "anantam vadda unna binduvulu" oka sthira bays point nundi praarambhamayye geodesic kiranaalanu (teachmueller metrik kosam) kaligi untaayi. eee sankshepana bays point empikapai aadharapadi untundi kabatti modular samuuham dwara charya teesukodu, vaastavaaniki kerkhoff eee sankshepanapai modular samuuham yokka charya nirantara charyaku vistarinchadani choopinchaadu.
thotamaali-masur sankshepana.
 thurston sankshepana maadirigaane sankshepanagaa pariganinchabadutundi, conny hyperbolic podavu kante teevra podavunu upayogistundi. modular group eee sankshepanapai nirantaram panichestundi, conny vaati sankshepana anantam vadda khachitamgaa ekkuva paayintlanu kaligi undani vaaru chuupimchaaru.
pedda ettuna jyaamiti.
teach mueller metrik kaligina teach mueller space yokka rekhaganita lakshanaalapai vistrutamaina adhyayanam jarigindi. telisina pedda-sthaayi lakshanaalu
marovaipu, tichmullar space hyperbolic spacela yokka aneka lakshanaalanu pradarsistundi, avih
eee lakshanaalalo konnintini hyperbolic anie piluvabade teach mueller space nundi karv complexes varaku myapla adhyayanam dwara vivarinchavacchu.
sanklishta jyaamiti.
bers embedding suutram _ 1kii suutram _ 183 yokka bahiranga upasamitigaa sanklishtamaina nirmaanaanni istundi.
sanklishta nirmaanam nundi vachhe kolamanalu.
teachmueller space oka sanklishtamaina aneka retlu kabatti idhi oka caratheodary metriknu kaligi untundi. teachmueller space anedi kobayashi hyperbolic mariyu daani kobayashi metrik teachmueller metrik thoo samanamga untundi. mapping class group anedi teach mueller metrik kosam isometryla puurti samuuham anie royden rujuvulo eee taruvaati phalitam upayoginchabadutundi.
bers embedding anedi techmullar spacenu holomorphy yokka domainga grahistundi, anduvalla idhi bergman metriknu kooda kaligi untundi.
techmullar sthalampai kahler kolamanalu.
veyil-petterson metrik anedi kahler conny adhi puurti kaadu.
techmullar sthalamlo oka pratyaekamaina puurti kahler-inestein metrik undani cheng mariyu yau chuupimchaaru. idhi sthiramaina pratikuula scalar vakratanu kaligi untundi.
techmullar space kooda kohler-hyperbolic dwara pravesapettabadina parimita vibhaga vakrata yokka puurti kahler metriknu kaligi untundi.
kolamanala samaanatvam.
asampoornamaina veyil-petterson metrik minaha, ikkada pravesapettina teachmueller sthalampai anni kolamanalu okadaanikokati pakshika-isometricga untaayi.

