Luận văn Đánh giá các phép biến hình á bảo giác những miền nội tiếp trong hình vành khăn

ChU'O11g2.DINH NGIDA PREP BIEN HINH K-A BAo GIAC vA MOT s6 CONGCD. . 2.1Mo duncuami~n hilien. B6 d~2.1.(T.Carleman[4,tr.212]) , , Gia sir w=fl(z) lamotPBHBGdandiep hinhvanhkhan~=K ~ lenmotmien. . b' b . nhilienG khongchuadi~m0Ci saochobientrongc cuaGtuangirngv6iduemg trimIzl=r. GQis la di~ntich(ngoai)cuat~pdongdoc baabQc,S la di~ndch (trong)cuat~pmadoC baabQc.Khi dotacoquailh~sau: s~(:)'s, (2.1) trongdodingthucxayrakhivachikhif(z)=az+bv6ia,b lacachfuIgs6va a:;tO. Tir b&dStasuyra cach~quasau: H~qua2.1.(Binb ngbiamodunmi~nnbi lien) N@umiSnnhilienG quacacPBHBGdandi~pf va fl IAnluQ'thanhhaihinh VaMkhan:H: r <Iwl< R va HI :rl <IWI1<RI . thi: R=RI f fl (2.2) DlililuangR duQ'cdinhnghlalamaduncuamiSnnhiliennoitrenkYhi~ular m(G)=R.r 7 Ron mranSu Gila anhcuaG b6imQtPBHBG dondi~pthi m(G1)=m(G), do la tinhb~tbiSncuama dunmiennhi lien. H~qua2.2(tinbdondi~uctiamodunmi~nnbilien) NSucacmiennhilienG vaG' v6icacmadunlftnluotR va R' cotinhch~t . . r r' Gc G'vaG ngfmcachhaithanhphftnbiencuaG'thi R::;;R' , D~ngthucxayr r' rakhivachikhi: G ==G' . 2.2 Binb ngbiapbepbi~nbinbK-a baGgiac Binb ngbia2.1 PhepbiSnhinhmQtmQtw =f(z) tirmienA trongm~tph~g z lenmienB trong m~tph~ngw'lien t\lChai chieuva baatoanchieuduongtrenbienduQ'cg9i la PBHKABGnSum9iill giaccongQc A comadunla m(Q)thiill giaccong Q'=f(Q)comadunlam(Q')seth6ab~td~ngthuc: ~m(Q)::;;m(Q')::;;Km(Q) (K ~1) K (2.3) SaildaylamQts6tinhch~tcuaphepbiSnhinhK-a baagiac: 1. K=1thiphepbiSnhinhtr6thanhbaagiac 2. RQ'PcuahaiphepbiSnhinhK}-a baagiacvaphepbiSnhinhK2-a baa giaclamQtphepbiSnhinhK}.K2-a baagiac. 3. PhepbiSnhinhnguQ'cuaphepbiSnhinhK-a baagiaclaphepbiSnhinh K-a baagiac. 8 4. NSuQ lahinhchfrnh~tABCDcocacc~ laa,b,Q' lahinhchfrnh~t A'B'C'D'co cacc~nha',b'songsongv6i cactf\lCto~dQ.Giasir[ la PBHKABGmi~nQ leuQ' saDchob6ndinhtuangUngv6iMall.Khi do: , :' =K: khivachikhi[cod~g[(x+iy)=a(Kx+iy)+p, ~ a' 1a kh' , h' kh . [ , d f( ' ) ( 1 ' ) A- =-- 1va c 1 1 co ang x +lY =a - x+lY +1-', b' Kb . K v6icach~ngs6a >0, f3phuc. SaudaylamQts6b~td~ngthucmarQngchoPBHKABG. B8d~2.2 GiasirG lami~nhilienv6imodunla m(G) duQ'cbiSnK-abaagiacleumi~n G' v6imodunm(G').Khi do: 1 K [m(G)JK~m(G')~[m(G)J . (2.4) Trang truanghQ'PG ={z,q<Izi<I} va G' ={w,q'<Iwl<I} thi (2.4)trathanh: 1 qK ~q'~qK . (2.4a) 1 1 Ran nfraq'=qK [( z) =aIzIK-lz, la!=1,va q =qK f (z) =aIzIK-l,lal=1. B8d~2.3(SoyrQngbAtdingthueCarlemanehoPBHKABG) Giasir[la PBHKABG hinhVaM khan(0<)r<Iz 1<R « 00)lenmi~nnhi lienG khongchuadiSm00 v6i bientrongc, bienngoaiC saDcho 1 z 1=R tuangUng 9 v6ibienngoaiC. GQiS la di~ntich(trong)cuat~pmadobienngoaiC bao bQc,sladi~ntich(ngoai)cuat~pdongdocbaobQc.Khi do: s~(~)%s, (2Ab) trongdodfuIgthucxiiyrakhivachikhi: ~~ ., w=f(z)=alzIK z+b v6i cachangso a:;t:Ovab. ChUngminh:(XemThao[14,tr.55-56]ho~cLuong[11,tr23]). B&d~2.4 (Bit ding thu-cdi~ntich cho mi~nda lien) Giii sir A la hinh vanh khan (0<)r <Iz 1<R (<00)v6i p nhat c~t Lj(j =O,I-,...,p-1) n~mtrendUOngtrOlldangtamI z1=Rl (r<Rl <R). GQiflit PBHKABG miSnA lenmiSnB cuam~tphfuIgw saochoduOngtrOllI z 1=R tuongUngv6i bienngoaiC cuaB, cacnhatc~tLj biSnthanhcacbien o)j =O,I,...,p-1). Khi dotaco: S(R, f) >S(r, f) ( ~ J Yrc +i> ( ~ J ~ r . J 'R 'J=1 1 (2.5) trongdo S(R,f) 1ftdi~ntich(trong)cuat~pmadobienngoaibaobQc,S(r,f) la di~ntich(ngofti)cuat~pdongdobientrongbaobQc,Sj la di~ntich(ngoai)cua t~pdongdo cac ITj baobQc. 1. --1. , Bangthucxiiyrakhi vachikhi fez)=a I z IK +b vaia,b lacachangso, a:;t:O. 10 ChUngminh:XemThao[14,tr.56]ho~cThao[13,tr.522]. 2.3 Caehamsapht}T(p,r,s)vaR(p,t,s) Gia su hiOOvaOOkhan r <Iz 1<1 tuangduangbaogiacv6i hiOOvaOOkhan s<Iw 1<1bi c~tp do~ndoctheobankiOO: Pj={wl s~lwl~t,argw=j21t},(j=O,1,...,p-l)(xem hiOO2.1), do tiOOdon p di~ucuamadunmiSnOOilien(h~qua2.2)taco 0~s~r <t <1.Han nuat la d~iluangxacdiOOduyoofittheor va s, r la m<)td~ilugngxacdiOOduyoofit theotvas. BG Or 1 1 z HiOO2.1(v6iP =2) Tir dotadiOOnghiacaehams6: t=T(p,r,s)v6ip=1,2,...;O~s<r<l, r=R(p,t,s) v6ip= 1,2,...;O~s<t<1. Clingdotinhdondi~ucuamadunmiSnnhi lientacocaetiOOchfitcuacaeham T(p,r,s) va R(p,t,s) OOusau: 1.r<T(p,r,s)<I(O~s<r<l) (2.6) 2.T(p,r,sl)>T(p,r,sz)(0~Sl<Sz<r <1) (2.7) 3.T(p,rps)<T(p,rz,s)(0~s<r1<rz<1) (2.8) (2.9)4. s<R(p,t,s)<t (O~s<t<l) 5.R(p,tpS)<R(p,tz,s)(0:5;t1<tz<1) (2.10) 6.R(P,t,SI)<R(p,t,sz)(O~SI<sz<t<l) (2.11) 7.R(p,t,s)>R(1,t,s)(0~s<t <1,p~2) (2.12) Nha caecongthuctrongNehari[10,tr.280-295],Thao[12,tr.100-107] va Luang[11,tr.15-18]diichiracongthuccuahamR vaT OOusau: R(P,t,s)=exp ( -JrKf(U» ) ,voiu=1+h-.Jh(2+h), trongd6: , 2pK(u) (l-k)(l-ak) 00 [ 1+s4pj ] h= ,k=4spfI " k(1+a) j=l 1+S4PJ-Zp ( i2pb t )a=sn b+ ;-In;,k ,b=K(k), 6 daysn(z,k)chisinelipticv6ithamsak. .!. 00 [ 1+r4pj ] ~ T (p,r,O)=4PrfI 4 '-2 (O<r<l,pEN) , j=l 1+r PJ P ( . a dx JT(p,r,s)=sexp2;:;k)!~(1-x')(1-k'x') , 0<s<r <1,pEN, voi K(k) 0011'tren 12 I-m k(l-h ) 2 00 [ l+r4pj ] a= m= h=4rP k +m' 2h(1- k)' U 1+R4pj-2p. Tir d6surracaet£OOeh~tkhaeeuahamR(p,t,s),T(p,r,s): -1 8. R(p,t,s)~4Ptkhit~0, (2.13) 1{2 9. l-R(p,t,s)~ 8 khi t~l 2pln p(1- t) (2.14) -1 10.4Pt<R(p,t,s)<t, O:5;s<t<l, \:fp, (2.15) 1 11.T(p,r,0)~4Pr,khi r~O (2.16) 8 ( 1{2 J 12.1- T(p,r,s)~ -exp khir ~ 1, P 2p(l- r) (2.17) 1 13.r<T(p,r,s)<4Pr,0:5;s<r<I,\:fp. (2.18) 2.4Cacb6d~khac: B6d~2.5(BAtding thtfcGrotzschl) Giasirtrongm~tph~ngz ehotruaemienEo ehotruae,giaih~ b6i Izl=1, Izl=q,O<q<1va p(l:5;p<oo)thanhphAnbien cr1,cr2,...,crpnb trong q<Izi<1. Gia sirhamw =f(z) bi~nbaagiacdandi~pmienEo lenmienBo ehuatrongmQthiOOvaOOkhanq'<Iwl<1 saoehoIzi=1ehuy~nthanhIwl=1, Izi=qehuy~nthanhIwl=q', crjthanhcrj'(j =1,2,...,n).N~ut~tea crj'laOOung 13 nhatc~theobankinhthitaviStfo,qothayvi f,q', trail~inSut~tca crj' la nhUngnhatc~theocaccungirond6ngtamt~i0 taviStfl,Qothayvi f,q'. GrOtzschdffchiraquanh~sail: < '<Qqo- q - 0, " (2.19a) trongdo q'=qof =fovaq'=Qof =fl . ChUngminh:xem[6,tr.372]. Bay gia nSumiSnEo noi trenduQ'chamw =<p(z)biSnK-a baagiaclenmiSn BI chuatrongq"<Iwl<1 saocho Izl=1 chuySnthanhIwl=I, Izi=q chuySn thanhIwl=q:'. Khi do I q~~q"<QK, (2.19b) trong do q"=q~<p(z)=<Po(fo(z)) v6i <Po(u) =alulK-Iu, lal=I, I I q" =QK <p(z) =<PI (fl (z)) v6i <PI(u) =alulK-I u, lal= 1. va ChUngminh:XemThao[14,tr.58]ho~cLuang[ll, tr.29]. BBd~2.6(Mo- rQngbAtding thuc Grotzsch2) GiasirD lahinhvanhkhanR <Iz1<1v6ipn(p=1,2,...,n=O,I,2...)nhatc~tn~m trencac duangd6ngtam 0 sao cho D trimgv6i chinhno b6i phepquay .21t 1-, '" Z=e P z, f la phepbienhinhK-a baagiacmienD lenmienEl namtrong 0<Iw 1<1sao cho duang iron 1 z 1=R tuanglIngv6i bientrongCI, saochot~p donggi6i h~ b6i CIchuag6ctQadQ,duangiron I z 1=1 tuanglIngv6ibien 14 .27t 1- ngoaiC2.Han111lagiil sirEl trimgv6'ichinhnoquaphepquayW =e P W .Khi dotacodanhgiadung M}::;T(p,RX ,m}), (2.20) v6'iM}=max{lwl,WEC}}, m}=min{lwl,wEC}}(~O),va T(p,r,s) 1ahamph\l du<;ycdinhnghTatrong2.3. . ~ Dangthucxilyraa (2.20)fez) =fo(z)=ah(t),1 a 1=1,t =biz IK z,1b 1=1, h 1a 1 phepbi@nhinhbilogiachinhvanhkhanRK <I t 1<1 1enmiSnnhilienP saocho } Itl=1 tuang'Ungv6'iIwl=1 va 1 t 1= RK tuangUngv6'ic trongdoc du<;ycdinh nghTanhusau: c={w,1w 1=m}}u {w,m}::;1w I::;Mpargw =j 21t},j=O, ,p-1. P ChUngminh:XemThao[14,tr.63],ho~cLuO'ng[19,tr.33]. DBd~2.7(M6'rQngbit dingthucGrotzsch2) Giil sir Dl 1ahinhvanhkhanQ<Izi<Rv6'ipn (p=I,2,...;n=0,1,2...)nhatc~t n&mtrencacduangtrOlld6ngtam0 saocho Dl trUngv6'ichinhno quaphep .2J1: quayZ =e1--;z, f 1aPBHKABG miSnD}1enmiSnEzn&mtrong0<Iwl<00 sao choduangironIzi=Q thanhbientrongC}baogBctQadQ,duangtrOllIzi=R thanhbien ngoai C2, Han nua E2 trimgv6'i chinhno quaphepquay .2J1: 1- W =e P w .Khi dotacodanhgiadUng 15 rnt ) 'rn> t 2 - Q K rnt T[P.(R) 'M, (2.21) rnj=min{lwllwECj},j =1,2va M2=rnax{lwllwECz}'v6i T(p,r,s)lahamph\! duQ'cdjnhnghi'atrong2.3. t D~ngthuc Kay ra ~f(z)=fo(z)=aH(t),lal=l,t=blzIK-tz,lbl=l,H la t t 1 PBHKABG hinhVaMkhanQK <It 1<RK tenmiennhjlienP saochoItI=RK tuangli'ngv6i c ={wllwl=M,}u {ill, ,;;1wI:,;M" argw=j 2; },j =O,...,p-1. It I =Qk wang t'mgv6i c ={wII w 1=rnt}. ChUngminh:B6 de2.7sur ill b6 de2.6nhacacphepbiSnd6i Z=Q vaz w =rnt, xernThao[14,tr.64]. w 2.5 Ly thuy~tdQditic1}'ctrj B8d~2.8 Trongrn~tphkg z chohinh chftnh~tD ={z=x +iy I 0 <x <a,O<y <b}. Giasirhams6 w =f1(z) th\lchi~nPBHKABG hinhchftnh~tD tenrnQtill giac congH cuarn~tphkg w saochocacdinh0,a,a+ibvaib cuaD lfu1lugttuang 16 tmgv6i cacdinh WI' W2' W3'va W4 cuaH. GQi r lahQcacclingytrongH n6i c~nhWIW2 v6i c~nhW3W4 cuaH. Gia sir co ham dQ do p=p(w)~0lien t\1c trong H sao cho O<lp(Y)=Ipldwl~oo,VYErva y 0 <Sp(H) =ffp2dudv<00,W=u +iv t6nt~itheonghiaLebesgue H D~t Ip=inf Ip(y) Khi dotaco Sp(H) ~~~l~ YEr K b (2.22) Chuy : DAngthuccothSxayra. ChUngminh: Trong truemghqpK =1 D~tcrx=Dn{zliRz=x}vayx=f(crx),O<x<a. TheogiathiSttaco a S p (H) = II p2dudv=II p2If;(z)12dxdy= I dx I p2If;(z)12Idyl.D ax H 0 TheobfitdAngthucSchwarzltanh~nduqcVx E(O,a) Lx P 2 If'(z)12 Idy I Lx Idy I ~ (Lx If'(z)lldyIf vado i Idyl=b >0 nencoax 2 Lx p2If' (z) 12IdY I ~ ~ (L xcrIf' (z) lidY I) dSY YxEr taco 17 1 a ( ) 2 1 a ( ) 2 Sp(H)~-f i plf'(Z)lldyldX=-f IpldWI dxb 0 crx b 0 Yx ~ ~12 afdX = ~12.b P b P0 (2.22a) TruemghQ'PK>1. Xet hI(w)1aPBHBG H 1enhinhchfrnh~tD' , v6i D' dugcdinhnghianhusau: D'={11=s+itlO<s<a',0<t <b'},hiSnnhienh1il 1aPBHKABG ill D 1enD' dodo a' 1 a->-- b'- K b. (2.22b) Apd\mg2.22achophepbiSnhinhh~1taco: a ' S p (H )~ b'1~ (2.22c) ThaykStquatfong(2.22b)vao(2.22c)tadugcSp(H) ~~: 1~. , Dfiu bimg co thS Kay fa tfong truemghQ'PH trimg vdi D', pew) =1,va : =K :' . khido~~12=~~(b,)2 =a'b'=S (H).KbP Kb P BBd~2.9 Trangm~tph~ngz chotu giac cong E ={zIfI 12(< 7t)} 18 Giasirhams6W=fez)thgchi~nPBHKABG miSnE IenmQtill ghiccongH cua matPhtmgw saDchocacdinhz =r ei<p2 Z =1",eicp\z =r eicp\z =r eiCP2cuaE. I 1'21'32'42 I~nluqt tuang(rugv6i cacdinh WI'w2'W3'W4cuaH. V6i cackyhi~up, Ip(Y),Y E r,Sp(H)nhu trongb6dS2.8taco: Sp(H)2 1 2-112K r p'In ---L rl (2.23) Chuy: f)~ngthuccothSxayfa. ChWzgminh: DungphepbiSnhinht=Inz biSnbaagiacmiSnE Ienhinhchunh~tD v6icac dinhtuang(rugsaudoapd\lngb6dS2.8tadugckStqua(2.23). B8d~2.10 Gia sir trongm~tph~ngz chotru6cmiSnA2 tuyY saDchov6i mQir ma (0 <)<r1<r <r2« 00)co 01<~(~ 2n), trong do <p=argzvaYr=A2 (\ {z:1z 1=r}. Yr Hams6w=f(z)thgchi~nmQtPBHBG dondi~pmiSnA2 IenmiSnB2cuam~t ph~ngw.Ta d~t: A21 =A2 n {z:rl<Iz 1<rJ va - B 21 =f (A 21)va y =f (y), rl < r <r2. Hanllliagiasir p=p(w) ~0 dugcxacdinhtrongB21saDcho 19 f P 2 dud v « 00 ), r1 < r < r2 y (0 <)Sp(B 21)=ffP2dudv « 00), w =u +iv, t6n t~i BZ1 Lebeguevav6imQir, r1< r < r2 c6 1p I dw I ~ 1p Yr . TZ d khi dota co: (1p ) 2 J ~ < sp (B 2 1 )T ra r)I ChUngminh:xem[12,tr 124-125]. 20 theo nghia (2.24)