Интегрируемость комплексных гамильтоновых систем с неполными потоками в C^2, страница 20

ï.á) íèæå) ïðè ïðîåêöèèà) êàæäàÿ ôóíêöèÿåéÿâëÿåòñÿ ìíîãîçíà÷íîé àíàëèòè÷åñêîé1TC2 ( dJdI1 (I1 )) := C/2π(Z ⊕òîð ñ ïàðàìåòðîìêó â ñëó÷àåε,1ÿâëÿåòñÿ ãîëîìîðôíîé ôóíêöè-áåç êðèòè÷åñêèõ òî÷åê, åå ìíîæåñòâîçíà÷åíèée ε,k := Ik (Uε (T0 )) = Ik (Gε,k ),Dîòêðûòî âCb ε,k := Jk (Uε (T0 )) = Jk (Gε,k ) ⊂ CDè ãîìåîìîðôíî îòêðûòîìó êðóãó, îíà âûðàæàåòñÿ â îêðåñò-130íîñòèUε (T0 )÷åðåç ëþáóþ äðóãóþ òàêóþ ôóíêöèþ ôîðìóëàìèIk = Ik (f (I` )),ãäåf (Ik )èñòâåííî (kf (Jk )Jk = Jk (f (I` )), ôóíêöèè, îáðàòíûå ê ôóíêöèÿìJk = Jk (f (J` ));Ik (f )èJk (f )ñîîòâåò-= 1, . .

. , n);á) ïðè ëþáûõk = 1, . . . , nèe ε,kIk ∈ Dϕk mod 2π|Gε,k ∩Tf (Ik )êîîðäèíàòû óãîëîáëàñòèIk = Ik (f (J` )),ìíîæåñòâî çíà÷åíèé êîìïëåêñíîéïîëó÷àåòñÿ èç íåêîòîðîé çàìêíóòîé(k)(k)Wk,Ik ⊂ C, îãðàíè÷åííîé øåñòèóãîëüíèêîì ñ âåðøèíàìè A1 (Ik ), . . . , A6 (Ik )(k)(k)C (âûðîæäàþùèìñÿ ïðè k = n â ïàðàëëåëîãðàìì ñ âåðøèíàìè A1 (Ik ), A2 (Ik ),(n)(n)(n)(n)A3 (Ik ) = A4 (Ik ), A5 (Ik ) = A6 (Ik ))ãåîäåçè÷åñêèìçè÷åñêèìè ñòîðîíàìè, ñîîòâåòñòâóþùèìèdk (f (Ik )), s2k−1 (f (Ik )), s2k−2 (f (Ik )) ⊂ Tf (Ik ) ,s2k (f (Ik )), s2k+1 (f (Ik )) ⊂ Tf (Ik )â ñëó÷àåk < n,à òàêæå ãåîäå-ñëåäóþùèì îáðà-çîì: (i) âûêèäûâàíèåì âñåõ âåðøèí (ñîîòâåòñòâóþùèõ áåñêîíå÷íî óäàëåííîé òî÷êåpf (Ik ) ),è (ii) îòîæäåñòâëåíèåì (ò.å.

ñêëåèâàíèåì) ïðè ïîìîùèïàðàëëåëüíîãî ïåðåíîñà ëþáîé ïàðû ñòîðîí, îòâå÷àþùèõ îäíîé è òîé æåãåîäåçè÷åñêîé (ëèáîdk (f (Ik )),ëèáîs1 (f (I1 ))ïðèóãîëüíèê (ñîîòâåòñòâåííî ïàðàëëåëîãðàìì ïðèk = 1);k = n)ñÿ ñëåäóþùèìè óñëîâèÿìè (ñì. ðèñ. 4.5, 4.6 ïðèïðè÷åì øåñòè-îäíîçíà÷íî çàäàåò-1 ≤ k < n, 1 ≤ k = nñîîòâåòñòâåííî):•øåñòèóãîëüíèê (èëè ïàðàëëåëîãðàìì)∂Wk,Ik ⊂ Cîáðàçîâàí òðåìÿ ïà-ðàìè ðàâíûõ è ïàðàëëåëüíûõ ñòîðîí, ñîîòâåòñòâóþùèõ ñëåäóþùèìãåîäåçè÷åñêèì è ïîëó÷àþùèõñÿ äðóã èç äðóãà ñëåäóþùèìè ñäâèãàìè âïëîñêîñòè(k)C:(k)2π(k)(k)A1 (Ik )A2 (Ik ) = ϕk (dk (ξ)) −→ A5 (Ik )A4 (Ik ) = ϕk (dk (ξ)),131(k)(k)A2 (Ik )A3 (Ik ) = ϕk (s2k−1 (ξ))(k)2,ξ := f (Ik ) ∈ D0,εδ∈Cíà âåêòîð−→â ïëîñêîñòèhDk (f (Ik ))ik := 2π•ïðè ëþáîìk<n(k)(k)A1 (Ik )A6 (Ik ) = ϕk (s2k−2 (ξ)),(k)(k)A6 (Ik )A5 (Ik ) = ϕk (s2k+1 (ξ)),δ:C→C÷åðåç−→−hDk (ξ)ik(k)A3 (Ik )A4 (Ik ) = ϕk (s2k (ξ))ãäå−hDk (ξ)ikk < n,îáîçíà÷åí ïàðàëëåëüíûé ïåðåíîñC, s0 (ξ) := s1 (ξ),dJk−1dJdJk1(Ik ) −(Ik ) + .

. . + (−1)k−1(Ik ) ,dIkdIkdIkâûïîëíåíî−−−−−−−−−−−→ −−−−−−−−−−−→(k)(k)(k)(k)A3 (Ik )A4 (Ik ) = A6 (Ik )A5 (Ik )dIk+1dIk+2n−k−1 dIn= hSk+1 (f (Ik ))ik := 2π(Ik ) −(Ik ) + . . . + (−1)(Ik ) ,dIkdIkdIk•òî÷êà ïåðåñå÷åíèÿ äèàãîíàëåé ïàðàëëåëîãðàììàðàâíà(k)12 (A1 (Ik )(k)(k)+ A3 (Ik )) = 0 = ϕk (0, a2k (f (Ik )))(k)(k)π = ϕk (0, a2k+1 (f (Ik ))),(1)(k)(k)A3 (Ik )A4 (Ik )A5 (Ik )A6 (Ik )(1)(1)à ïðèk=1ðàâíà(k)(k)(îòêóäà òî÷êà ïåðå-ñå÷åíèÿ äèàãîíàëåé (âûðîæäàþùåãîñÿ â îòðåçîê ïðèëîãðàììà(k)A1 (Ik )A2 (Ik )A3 (Ik )A6 (Ik )k = n)(k)12 (A3 (Ik )ïàðàëëå-(k)+ A5 (Ik )) =öåíòðû îòîæäåñòâëÿåìûõ ñòîðîí(1)A2 (I1 )A3 (I1 ) è A1 (I1 )A6 (I1 ) ñóòü îòîæäåñòâëÿåìûå òî÷êè ± 21 hD1 (f (I1 ))i1 =ϕ1 (0, a1 (f (I1 ))));â ÷àñòíîñòè, ïðèêîîðäèíàòûk=1äëÿ ëþáîãîe ε,1I1 ∈ D1ϕ1 mod 2π|Gε,1 ∩Tf (I1 ) : Gε,1 ∩ Tf (I1 ) → (TC2 ( dJdI1 (I1 ))) \ γI1âåñü ïîïîëíåííûé íàäðåçàííûé òîððîìîáðàçîì êîìïëåêñíîé óãëîâîé1TC2 ( dJdI1 (I1 ))â ñëó÷àån = 1)1(TC2 ( dJdI1 (I1 ))) \ γI1(ñîâïàäàþùèé ñ òî-çà èñêëþ÷åíèåì äâóõ òî÷åê(ñîâïàäàþùèõ äðóã ñ äðóãîì â ñëó÷àån = 1),(1)(1)A3 (I1 ), A4 (I1 )ÿâëÿþùèõñÿ êîíöàìè ëèíèèíàäðåçà è îòâå÷àþùèõ áåñêîíå÷íî óäàëåííîé òî÷êå;132ÿâëÿåòñÿâ)(dz ∧ dw)|Gε,k = dIk ∧ dϕk , k = 1, .

. . , n;ã) ïåðåìåííàÿ äåéñòâèåIk (ξ) =1πIk = Ik (f )è ôóíêöèÿa2k+1Z (ξ)pξ − P2n+1 (y)dy, Jk (ξ) =a2k (ξ)aZ2k (ξ)1πJk = Jk (f )èìåþò âèäpξ − P2n+1 (y)dy,2ξ ∈ D0,ε,a2k−1 (ξ)q√ãäå â êà÷åñòâå ôóíêöèèáåðóòñÿ åå âåòâè, òàêèå ÷òî−P2n+1 ( 12 (a2k + a2k+1 )) >q0 â ïåðâîì ñëó÷àå, è i −P2n+1 ( 21 (a2k−1 + a2k )) < 0 âî âòîðîì ñëó÷àå;ä) äëÿ ëþáûõ äâóõ ðó÷åêGε,k , Gε,` ,òî æå ñåìåéñòâî ãåîäåçè÷åñêèõ1≤k<nïåðåñå÷åíèåGε,k ∩ Gε,k+1 =sj (ξ),Gε,k ∩ Gε,k+1[ñîäåðæàùèõ â ñâîåé ãðàíèöå îäíî èk = ` ± 1,âûïîëíåíîïðè÷åì â ñëó÷àåÿâëÿåòñÿ îáúåäèíåíèåì ãåîäåçè÷åñêèõ(s2k (ξ) ∪ s2k+1 (ξ)) =|ξ|<ε[(Prw |Tξ )−1 (Sk+1 (ξ)),|ξ|<εè íà ýòîì ïåðåñå÷åíèè êîìïëåêñíûå êîîðäèíàòû óãîëϕk mod 2π è ϕk+1 mod 2πñâÿçàíû äðóã ñ äðóãîì ôîðìóëàìè:00Ik+1(ξ) ϕk+1 |s2k (ξ) + πJk+1(ξ) = Ik0 (ξ) (ϕk |s2k (ξ) − π),00Ik+1(ξ) ϕk+1 |s2k+1 (ξ) − πJk+1(ξ) = Ik0 (ξ) (ϕk |s2k+1 (ξ) − π);å) óðàâíåíèÿ Ãàìèëüòîíà â êîîðäèíàòàõk = 1, . .

. , n,(Ik , ϕk mod 2π)íà ðó÷êåïðèíèìàþò âèä:I˙k = 0,4) àíòèêàíîíè÷åñêàÿ èíâîëþöèÿùàÿ Ãàìèëüòîíèàíϕ̇k =df (Ik );dIkC2 → C2 , (z, w) 7→ (−z, w),ñîõðàíÿþ-f , ïåðåâîäèò êàæäóþ ÷åòûðåõìåðíóþ ε-ðó÷êó Gε,káÿ, è îãðàíè÷åíèå ýòîé èíâîëþöèè íà ýòó ðó÷êó â êîîðäèíàòàõèìååò âèäGε,k ,(Ik , ϕk mod 2π) 7→ (Ik , −ϕk mod 2π), 1 ≤ k ≤ n.133â ñå-(Ik , ϕk mod 2π)Ðèñ. 4.5: Øåñòèóãîëüíèê ∂Wk,Ik ⊂ Cϕk ïðèÐèñ. 4.6: Ïàðàëëåëîãðàìì ∂Wn,In ⊂ Cϕn1≤k<nÄîêàçàòåëüñòâî.Ïóíêò 1) ñëåäóåò èç ñëåäñòâèÿ 4.1.3.

Ïóíêò 2) ñëåäóåò èç÷àñòíîãî ñëó÷àÿ ëåììû 3.3.1: êîãäà íà ñëîå T0 íåò îñîáûõ òî÷åê. Äîêàæåìïóíêò 3).Øàã 1. Îïðåäåëèì ôóíêöèè Ik = Ik (ξ), Jk = Jk (ξ) ôîðìóëàìè ïóíêòà ã).Äîêàæåì ïóíêò à), ò.å. ãîëîìîðôíîñòü ôóíêöèé Ik (ξ), Jk (ξ). Ïî ïîñòðîåíèþçíà÷åíèå ïåðåìåííîé äåéñòâèå Ik çàâèñèò òîëüêî îò ξ . Äàëåå,1dIk (ξ)= dξπa2k+1Z (ξ)ppd( ξ − P2n+1 (w))dw + a02k+1 (ξ) ξ − P2n+1 (a2k+1 (ξ))dξa2k (ξ)p1− a02k (ξ) ξ − P2n+1 (a2k (ξ)) =2πa2k+1Z (ξ)a2k (ξ)dwp.ξ − P2n+1 (w)Îòñþäà ñëåäóåò, ÷òî ñóùåñòâóåò ε > 0, òàêîå ÷òî ïðè |ξ| < ε ïðîèçâîäíàÿdIk (ξ)/dξ ñóùåñòâóåò, ïîýòîìó Ik = Ik (ξ) ÿâëÿåòñÿ ãîëîìîðôíîé ôóíêöèåé.2Áîëåå òîãî, ïîñêîëüêó dIk (0)/dξ 6= 0, òî ìîæíî ñ÷èòàòü, ÷òî Ik |D0,εÿâëÿåò-2e ε,k :=ñÿ äèôôåîìîðôèçìîì îòêðûòîãî êðóãà D0,εðàäèóñà ε íà îáëàñòü D2Ik (D0,ε).Àíàëîãè÷íîJk0 (ξ)=12πa2kR(ξ)√a2k−1 (ξ)134dw.ξ−P2n+1 (w)Ïóíêò à) äîêàçàí.Øàã 2.

Äëÿ ëþáîé êóñî÷íî ãëàäêîé îðèåíòèðîâàííîé êðèâîé αξ íà ñëîåTξ îáîçíà÷èìZhαξ i :=∆ξ ,hαξ ik :=αξhαξ i,Ik0 (ξ)(4.1.1)2ξ ∈ D0,ε,ãäå ∆ξ ãîëîìîðôíàÿ 1-ôîðìà íà ñëîå Tξ , ñì. îïðåäåëåíèå 3.1.11(À). Òàêêàê ôîðìà ∆ξ çàìêíóòà, òî åå èíòåãðàë hαξ i ïî ëþáîìó îðèåíòèðîâàííîìóöèêëó αξ ⊂ Tξ íå ìåíÿåòñÿ ïðè ëþáûõ ãîìîòîïèÿõ öèêëà â ñëîå.

Ðàññìîòðèìíåïðåðûâíûå ïî ξ ñåìåéñòâà îðèåíòèðîâàííûõ öèêëîâαξ,k := (Prw |Tξ )−1 (γa2k (ξ),a2k+1 (ξ) ) ⊂ Tξ ∩ Gε,k ,α̂ξ,k := (Prw |Tξ )−1 (γa2k−1 (ξ),a2k (ξ) ) ⊂ Tξñ òàêîé îðèåíòàöèåé, ÷òî α0,k è α̂0,k ÿâëÿþòñÿ èíòåãðàëüíûìè òðàåêòîðèÿìèïîëåé sgrad C f è i sgrad C f ñîîòâåòñòâåííî, k = 1, . . . , n. Òîãäàhαξ,k i = 2πIk0 (ξ),hα̂ξ,k i = 2πJk0 (ξ)(4.1.2)â ñèëó ôîðìóë äëÿ Ik0 (ξ) è Jk0 (ξ), ñì. øàã 1. Ïîýòîìó hαξ,k ik = 2π è ñóùåñòâóåòåäèíñòâåííàÿ ôóíêöèÿ ϕk mod 2π : Gε,k → C/2πZ ïðè 2 ≤ k ≤ n, ϕ1 mod 2π :Gε,1 \Ss1 (ξ) → C/2πZ ïðè k = 1, òàêàÿ ÷òî|ξ|<εd(ϕk |Gε,k ∩Tξ \s1 (ξ) ) = ∆ξ /Ik0 (ξ),2ξ ∈ D0,ε.ϕk mod 2π(0, a2k (ξ)) = 0 mod 2π,Ïóíêò â) ñëåäóåò èç òîãî, ÷òî dIk ∧dϕk = Ik0 (f (z, w))df (z, w)∧ 2zI 0 (fdw(z,w)) |Gε,k =k2zdz∧dw|Gε,k2z= (dz ∧ dw)|Gε,k . Ïóíêò å) ñëåäóåò èç ïóíêòîâ à) è â).Øàã 3.

Äîêàæåì ïóíêò á). Ñîãëàñíî ñëåäñòâèþ 4.1.8, ñóùåñòâóþò ãåîäåçè÷åñêèå s1 (ξ), . . . , s2n−1 (ξ) : (0, 1) → Tξ ðèìàíîâîé ìåòðèêè ds2ξ íà Tξ ⊂fε4 ñ êîíöàìè lim si (ξ)(t) = lim si (ξ)(t) = pξ ∈ Tξ , 1 ≤ i ≤ 2n − 1,Tξ ⊂ Mt→0+t→1−135Ðèñ. 4.7: Íàäðåçàííûé èëè ïðî- Ðèñ.

4.8: Îáëàñòüöèëèíäð Ðèñ. 4.9: Îáëàñòüöèëèíäðêîëîòûé òîð c1 (ξ) ∪ s1 (ξ) ⊂ Tξck (ξ) ⊂ Tξ ïðè 1 < k < ncn (ξ) ⊂ Tξfε4 áëèçêè ê çàìûêàíèÿì èíòåãðàëüíûõ òðàåêòîðèéçàìûêàíèÿ êîòîðûõ â Ms1 , . . . , s2n−1 âåêòîðíîãî ïîëÿ sgrad C f . Àíàëîãè÷íî, ñóùåñòâóþò ãåîäåçè÷åñêèå d1 (ξ), . . . , dn (ξ) íà (Tξ , ds2ξ ), çàìûêàíèÿ êîòîðûõ áëèçêè ê çàìûêàíèÿìèíòåãðàëüíûõ òðàåêòîðèé d1 , . . . , dn âåêòîðíîãî ïîëÿ i sgrad C f . Ïîýòîìó óêàçàííûå ãåîäåçè÷åñêèå íà ñëîå Tξ , èõ îáðàçû â C ïðè ïðîåêöèè Prw |Tξ : Tξ → Cè öèëèíðè÷åñêèå îáëàñòè ck (ξ), 1 ≤ k ≤ n, âûãëÿäÿò êàê íà ðèñ. 4.7, 4.8, 4.9.Ðàçðåæåì öèëèíäðè÷åñêóþ îáëàñòü ck (ξ) ⊂ Tξ (ñì.

îáîçíà÷åíèå 4.1.9) ïîãåîäåçè÷åñêîé dk (ξ) è ðàññìîòðèì îáðàç ïîëó÷åííîé îäíîñâÿçíîé (ðàçðåçàííîé) îáëàñòè ck (ξ)\dk (ξ) ïðè îòîáðàæåíèè ϕk , ÿâëÿþùåìñÿ âåòâüþ îòîáðàæåíèÿ ϕk mod 2π , ââåäåííîãî íà øàãå 2. Òàê êàê ãðàíèöà ðàçðåçàííîé îáëàñòèñîñòàâëåíà èç ãåîäåçè÷åñêèõ: dk (ξ) ∪ s2k−2 (ξ) ∪ s2k−1 (ξ) ∪ s2k (ξ) ∪ s2k+1 (ξ), 1 ≤ k < n,∂(ck (ξ)\dk (ξ)) = d (ξ) ∪ s1 ≤ k = n,k2n−2 (ξ) ∪ s2n−1 (ξ),ãäå s0 (ξ) := s1 (ξ), òî ãðàíèöà åå îáðàçà ϕk (ck (ξ)\dk (ξ)) ⊂ C ñîñòîèò èç ïðÿìîëèíåéíûõ îòðåçêîâ â ïëîñêîñòè Cϕk . Íåòðóäíî ïîêàçûâàåòñÿ, ÷òî êîìïëåêñíîçíà÷íàÿ ôóíêöèÿ ϕk |ck (ξ)\dk (ξ) : ck (ξ)\dk (ξ) → C èíúåêòèâíà, ïîýòîìó åå îáðàçÿâëÿåòñÿ âíóòðåííîñòüþ íåêîòîðîãî 6-óãîëüíèêà (ïðè k < n) èëè 4-óãîëüíèêà136(ïðè k = n)(k)(k)(k)(k)(k)(k)∂Wk,Ik = A1 (Ik )A2 (Ik )A3 (Ik )A4 (Ik )A5 (Ik )A6 (Ik )â ïëîñêîñòè Cϕk , ãäå Ik := Ik (ξ).

Çíà÷èò, ìíîæåñòâî çíà÷åíèé êîìïëåêñíîéêîîðäèíàòû óãîë ϕk |ck (ξ)\dk (ξ) : ck (ξ) \ dk (ξ) → C íà ðàçðåçàííîé öèëèíäðè÷åñêîé îáëàñòè ck (ξ)\dk (ξ) ÿâëÿåòñÿ âíóòðåííîñòüþ ýòîãî øåñòèóãîëüíèêà. Ïîýòîìó ìíîæåñòâî çíà÷åíèé êîìïëåêñíîé êîîðäèíàòû óãîë ϕk mod 2π|Gε,k ∩Tξíà çàìûêàíèè ck (ξ) = Gε,k ∩ Tξ ýòîé îáëàñòè ïîëó÷àåòñÿ èç çàìêíóòîé îáëàñòè Wk,Ik , îãðàíè÷åííîé ýòèì øåñòèóãîëüíèêîì, âûêèäûâàíèåì âñåõ âåðøèíè îòîæäåñòâëåíèåì ïàð ñòîðîí, îòâå÷àþùèõ îäíîé è òîé æå ãåîäåçè÷åñêîé.Âû÷èñëèì íàïðàâëÿþùèå âåêòîðà ñòîðîí øåñòèóãîëüíèêà ∂Wk,Ik :(k)(k)(k)(k)A2 (Ik ) − A1 (Ik ) = A4 (Ik ) − A5 (Ik ) = hdk (ξ)ik =: hDk (ξ)ik ,(k)(k)(k)(k)(k)(k)(k)(k)A3 (Ik ) − A2 (Ik ) = A6 (Ik ) − A1 (Ik ) = hs2k−2 (ξ)ik = hs2k−1 (ξ)ik =: hSk (ξ)ik ,A4 (Ik ) − A3 (Ik ) = A5 (Ik ) − A6 (Ik ) = hs2k (ξ)ik = hs2k+1 (ξ)ik =: hSk+1 (ξ)ik ,ãäå ïîñëåäíÿÿ öåïî÷êà ðàâåíñòâ âûïîëíåíà ïðè k < n.

Îñòàëîñü çàìåòèòü,÷òîhSn (ξ)i = hαξ,n i = 2πIn0 (ξ),hSk (ξ)i+hSk+1 (ξ)i = hαξ,k i = 2πIk0 (ξ), 1 ≤ k < n,hD1 (ξ)i = hα̂ξ,1 i = 2πJ10 (ξ),hDk−1 (ξ)i+hDk (ξ)i = hα̂ξ,k i = 2πJk0 (ξ), 1 < k ≤ nâ ñèëó (4.2.2). Ñ ó÷åòîì (4.2.1), îòñþäà ïîëó÷àåì òðåáóåìûå ðàâåíñòâà00Ik+1(ξ) − Ik+2(ξ) + . . .