
aalfozhs sidhaantham

nevenlinna sidhaanthathinte jyaamitheeya prathiroopamaayi larse aalfozhs kandupidicha oru ganithashaasthra sidhaanthamaanu aalfozhs sidhaantham. 1936ll eee sidhaanthathinu aadyathe rund feeld medalukalil onnu aalfozhsinu labhichu.
bhoopadangal ulkkollunnathinte adisthaana savisheshathakalude saamaanyavalkkaranamaayi ithu kanakkakkaam.
chila nannayi nirvachikkapetta ardhathil "eathaandu coveringukal" ulla maappukal. konformal remanian alavukal sajjeekarichirikkunna athirthi riman prathalangalkk ithu baadhakamaan.
praadhamika.
oru "boarder reeman uparithalam" "x" ennathu oru compact reeman uparithalathile oru pradeshamaayi nirvachikkaam, athinte athirthi shnam "x" parimithamaaya niravadhi disjoyint jordan vakrangal ulkkollunnu. mikka prayogangalilum eee valavukal vishakalanaparamaan, ennaal sidhaantham pravarthippikkunnathinni aavashyamaaya eee valavukalil chila vyakthamaaya kuranja kramamaya vyavasthakalundu; ithine "aalfozhs regularity" ennu vilikkunnu. konformal remanian metrik" ennathine nirvachichirikkunnathu "ds" enna neelamulla moolakamaan, ithu konformal local kordinettukalil "z" ennu "ds" = "ρ" ("z") aayi prakadippikkunnu. |"dii. isad"|ivide "ρ" ennathu otappetta poojyangalulla oru minusamarnna positteev functionaanu.
poojyangal illengil, metrik smooth ennu vilikkunnu. neelam ghadakam soothravaakyangal upayogichu sariyaakkaavunna valavukaludeyum pradeshangaludeyum neelam nirvachikkunnu.
appol rund pooyintukal thammilulla dooram vakrangalude neelatthinte ananthathayaayi nirvachikkappedunnu.
eee pooyintukal bandhippikkuka.
krameekaranavum notteshanum.
x", "y" enniva rund athirthi riman prathalangalayirikkatte, koodaathe "y" oru minusamarnna (athirthi ulppede) konformal metrik "σ" ("z") "dz" kondu sajjeekarichirikkunnave karuthuka. x" muthal "y" vareyulla oru holomorphic bhoopadamaayirikkatte "f". appol "x" ll "pull-back" metrik nilavilundu, athu nirvachichirikkunnathu
x" eee metrik kondu sajjeekarikkumbol, "f" oru "local isometry" aayi maarunnu; athaayath, oru vakrathinte neelam athinte chithratthinte neelathinu thulyamaanu. x", "y" ennivayile ellah neelangalum pradeshangalum eee rund alavukalumayi bandhappett alakkunnu.
f" "x" nte athirthi "y" nte athirthiyilekku ayaykkukayaanenkil, "f" oru "ramified covering" aanu. prathyekichum,
ippol "x"-nte athirthiyude chila bhagam "y"-yude ulbhaagathekku mapp cheythittundennu karuthuka. eee bhaagathe "aapekshika athirthi" ennu vilikkunnu. l" ennathu eee aapekshika athirthiyude neelamaayirikkatte.
aadyathe pradhana sidhaantham.
sharashari covering nambar nirvachikkunnath formulayanu
eee samkhya oru kavarinte digriyude saamaanyavalkkaranamaanu.
athupole, oro saadhaarana vakrathinum "γ" num "y" leeii oro saadhaarana pradeshathinum "d" num"
sharashari covering nambarukal nirvachikkappedunnu
aadyathe pradhana sidhaantham parayunnath oro saadhaarana pradeshathinum oro saadhaarana vakrathinum,
ivide "l" ennathu aapekshika athirthiyude neelavum "k" ennathu athine maathram aashrayichirikkunna sthiraankavumaanu.
"y," "σ", "d", "γ", ennaal "" f "," "" x "" ennivayil ninnu swathanthramaanu ".
l" = 0 aakumbol eee asamathvangal coveringukalude (a) property durbalamaaya analogaayi maarunnu.
randaamathe pradhana sidhaantham.
ρ" ennathu euler swabhaavathinte "negatteevu" ayirikkatte (athinaal "m" dwaarangalulla golatthinu "ρ" = "2m" −2). appol
ρ" ("y")> 0 aayirikkumbol maathrame ithu ardhavathaakoo, udaaharanathinu "y" moonnu (allengil athil kooduthal) dwaarangalulla oru golamaakumbol. eee saahacharyathil, bii) coveringukalude property saamaanyavalkkaranamaayi phalam kanakkakkaam.
aaplikkeshanukal.
ippol "z" oru ooppan reeman uparithalamaanennu karuthuka, udaaharanathinu complexes playin allengil yoonit disk, koodaathe "z" oru konformal metrik "ds" kondu sajjeekarikkappette. z" ll adangiyirikkunna athirthi uparithalangalude "d" "j" avayude adavukalodoppam, "z" ll samyojippichirikkunna, avayude koottukettukalum undengil ("z", "ds") "pathivayi theernnupokunna" ennu nammal parayunnu.
ekapaksheeyamaaya" konformal metrik ulla sankeernamaaya thalam pathivayi theernnupokumennu aalfozhs theliyichu. eee vastutha, rund pradhana sidhaanthangalumaayi chernnu picardinte sidhaanthathe soochippikkunnu,
nevenlinna sidhaanthathinte randaamathe pradhana sidhaantham. picardinte mattu niravadhi pradhana saamaanyavalkkaranangal
aalfozhs sidhaanthathil ninnu sidhaantham nedaanaakum.
prathyekichu sradheyamaaya oru phalam (andrey bloach neratthe oohicha) "anchu dweep sidhaantham" aanu.
anchu dveepukalude sidhaantham.
dii" 1 aakatte..dii" 5 ennathu riman golatthile anchu jordan pradeshangalaayirikkanam. appol eee pradeshangale maathram aashrayich inipparayunna savisheshathakalulla oru sthiramaaya "sii" nilavilunda
golaakrithiyilulla derivative" thripthippeduthunna vidhathil yoonit diskile oru meromorphic functionayirikkatte "f".
appol yoonit diskill adaykkappettirikkunna lalithamaayi bandhippichirikkunna oru mekhala "jii" undu, athaayath
f" ennathu "d" "j" pradeshangalilonnil "g" ennu homomorphically mapp cheiyunnu.
ithu naalu mekhalakalumaayi bandhappettathalla. udaaharanathinu, "f" ("z") = ′ ("kz") edukkuka, ivide "k"> 0 ekapaksheeyamaayi valuthaanu, koodaathe "′" ennathu deferential samavaakyam thripthippeduthunna warestrass eliptik functionaanu.
e" 1 "," e "2", "e" 3 ", φ ennee naalu pooyintukalude ellah preeimejukalum onniladhikam aanu, athinaal eee poyintukalkku chuttum disjoyint klosharukalulla naalu discukal edukkukayaanengil, eee discukalilonnum homomorphically mapp cheytha oru mekhalayum undaakilla.
paraamarshangal.
aalfozhsinte yadhaartha jenal pepparinu purame,
eee sidhaantham pusthakangalil vishadeekarichittundu.
randaamathe pradhana sidhaanthathinte lalithamaaya thelivukal eee prabandhangalil kaanaam
tokki
oppam dii thelin.
aalfozhsinte sidhaanthathe aashrayikkaatha anchu dweep sidhaanthathinte lalithamaaya oru thelivu bergweeler vikasippicheduthu.