Intersections of totally real and holomorphic disks

Publicacions Ma ema iques, Vol 37 (1993), 111-126 . A bs ac INTERSECTIONS OF TOTALLY REAL AND HOLOMORPHIC DISKS TOM DUCHAMP AND FRANC FORSTNERIC I is shown ha a holomo phically embedded open disk in (C2 and a o ally eal embedded open disk which ha e a common smoo h bounda y ha e non i ial in e sec ion . 1 . In oduc ion I is now clea , h ough he wo k o G omo and o he s, ha he e is a s ong ela ionship be ween he heo y o Lag angian imme sions in o symplec ic mani olds and he heo y o o ally eal imme sions in o complex mani olds . The ela ion ollows om he ac ha he G ass- mann space o Lag angian subspaces o R 2 n is homo opy equi alen o he G assmann space o o ally eal subspaces o C' . The e a e su p ising di e ences be ween he wo heo ies, howe e . Recall ha complex 2-space, (C 2 , is a symplec ic mani old wi h symplec- ic o m w = 2 (dz l A dz l + dz 2 A dz 2 ) . Le 0 : 0 ---> (C 2 be a symplec ic embedding o he open uni disk in (C, ha is, suppose ha he pull-back O*w is a symplec ic o m on A . I is no di icul o p o e ha e e y holomo phic embedding o 0 is symplec ic . Assume u he ha ~b ex ends o a smoo h embedding o he closed disk 0 . The e is no Lag angian embedding o 0 whose bounda y coincides wi h 0(0A) . Fo suppose ha 0 is such an embedding . Then O( 0 ) and 0(,i1) a e homologous ela i e o he bounda y ; hence, The second au ho was pa ially suppo ed by he Resea ch Council o he Republic o Slo enia . 11 2  T . DUCHAMP, F . FORSTNERIC which is a con ac ion because he i s in eg al is 0 and he second is posi i e . On he o he hand, i is easy o cons uc examples o a pai consis ing o a holomo phically embedded disk and a o ally eal disk which ha e a common smoo h bounda y (see Sec ion 4 below o one such example) . In his pape he in e sec ion heo y o such a pai is in es iga ed . The main esul is he ollowing . In e sec ion Theo em . I a holomo phically embedded disk and a o ally eal embedded disk in (C Z ha e a common bounda y hen hey in e sec in a leas one in e io poin . Mo e p ecisely, le 0 : 0 -> (C2 be a smoo h embedding o he elosed uni disk, holomo phic on A ; and je 0 : 0 ~___> C 2 be a smoo h o ally eal embedding such ha 0(ao) = 0(ao) . Then he in e sec ion 0(0) 1 O(0) is non-emp y . The p oo o he heo em is by con adic ion : Suppose he heo em is alse . Then he e a e embeddings 0 and 0 o 0 wi h o(0) 1 o(0) = 0 and 0(8 0 ) = O(aA) . We will show how o de o m he images O( 0 ) and 0( 0 ) o p oduce wo o ally eal embeddings which abu smoo hly along he bounda y O( 80 ) in such a way ha he union o he images o he de o med o ally eal embeddings de ines a o ally eal embedding o he wo sphe e in o C2 . Bishop [1] (see also Wells [5]) p o ed ha he e is no o ally eal embedding o he wo sphe e in o (C 2 . Acknowledgmen . We wish o hank P o esso E . L . S ou o b ing- ing his p oblem o ou a en ion . 2 . P ope ies o o ally eal embeddings Be o e beginning he p oo i is necessa y o e iew se e al condi ions which a e equi alen o he o al eali y condi ion and o de ine an index o o ally eal embeddings o annuli . 2 .1 . Condi ions o o al eali y . An embedding ip : U -> (C2, U C cC a egion, is said o be o ally eal i o all ~ E U he in e sec ion 0,T C UnJO * T C U is i ial . (He e TU deno es he eal angen bundle o U and J he complex s uc u e enso o C 2 .) Le (z, w) deno e complex coo dina es on C 2 and (u, ) a bi a y eal coo dina es on U . The nex lemma gi es se e al cha ac e iza ions o he o al eali y condi ion . They a e mo e o less well-known and easily e i ied, so we lea e i o he eade o check hem . (See S ou -Zame [4] o a discussion o o al eali y) . TOTALLY REAL AND HOLOMORPHIC DISKS  113 Lemma 1 . Le 0 : U - (C 2 be an embedding, U C (C, de ined by he unc ions z = Z(S), w= W«) . Then he ollowing a e equi alen : (i) The embedding 0 is o ally eal . (ii) Fo each ( E U he equali y (C - 0,,(T S U) = TC 2 holds l . az OZ/a~ (iii) The de e minan  Y  ne e anishes . aW/a( az az (i ) The de e minan  au  a  ne e anishes . aW aW au a Rema k 1 . Condi ion (ii) can be es a ed as ollows . Le X = (X 1 , X 2 ) be a basis o he angen space TAU, ( E U . Then he ec o s ~b* (X1) and 0* (X2) o m acomplex basis o he complex ec o space T, G(C)C 2 . We in oduce he no a ion A~  1 - a < l~l < 1} and Aá - {~ : 1_< l~i ~i <_ 1 + a} o a > 0 . The special case in which ~b is de ined on he annula egion AQ , a > 0 and 0 is o he o m «)=0 o ~(j=1 is o pa icula in e es o us . Fi s obse e ha condi ion (iii) educes o a~ o . I will p o e use ul o w i e equa ion (1) in pola coo dina es, z = eie, (2)  00 - i Y  0 . Because (z) anishes o all 1 z~ = 1, i ollows ha áe = 0and o al eali y implies ha - =,b 0 o all z wi h Iz1 = 1 . This implies he ollowing lemma . Lemma 2 . Le 0 be a o ally eal embedding o he annulus A as gi en abo e . Then he e is an annula egion o he o m 1 <_ Iz1 < 1 + a', 0 < a' < a < 1, on which can we w i en in he o m . (z) = R(z)e¡8(z) 'I V C W is a eal subspace o he complex ec o space W hen (C - V deno es he complex subspace spanned by V . 11 4  T . DUCHAMP, F . FORSTNERIC whe e R(z) is a non-nega i e, smoo h eal- alued unc ion which anishes o ~z1 =_1 and whe e O(z) is smoo hmodulo 27 . The inequali y  8 a í,(z) > 0 is sa is ied o all z in a neighbo hood o he uni ci cle Iz1 = 1 . Now suppose ha is any smoo h unc ion de ined on A o he o m . (z) = R(z)eio(z) Subs i u ion o he o mulas _ a  _  aR(z  a0 _ io  a  aR(z)  0" a - (  a  ) + iR(z) a ) e  and ó0 = (  00  + iR(z) a0 ) ego in o he o al eali y condi ion (2) and sepa a ing eal and imagina y pa s o he coe icien o e io yields he condi ion '9R  + R(z) ae a0 ) + i (R(z) 50 - a a z))  0 . In pa icula , i he imagina y pa o he le hand side is nega i e, he embedding is necessa ily o ally eal . Lemma 3 . Le 0 : Aá --> C2, 0 < a < 1, be an embedding o he o m O(z) = (z, R(z)eio(z)), whe e R and O a e eal- alued unc ions wi h R smoo h and O smoo hmodulo 27 . I he inequali y aR(z) > R(z)a0 a  á0 is sa is ied hen he embedding is o ally eal . 2 .2 . An index o o ally eal embeddings o annuli . In his sec ion we de ine an index o a o ally eal embedding o an annulus in (C 2 . I is closely ela ed o he Maslo index and is a special case o an index de ined by Kambe and Tondeu [3] . A de ailed p esen a ion, wi hin he con ex o o ally eal embeddings o su aces in C 2 , is gi en in [2] . We gi e a sel -con ained exposi ion he e . Le 00 : A -> (C 2 be any o ally eal embedding o an annula egion A C C . To de ine he index o 00 begin by choosing a complex aming 2 2 By a complex aming we mean a pai o complex ec o ields which a e poin wise independen o e (C . = ( ,, 2) o he holomo phic angen bundle T( 1 , 0 )U, whe e L is any open con ac ible neighhbo hood o A . Nex choose a eal aming 3 X = (X1, X2) o he eal angen bundle TA which is compa ible wi h he o ien a ion o A as a subse o (C . By i ue o Rema k 1, he e is a smoo h ma ix- alued unc ion de ined by he o mula TOTALLY REAL AND HOLOMORPHIC DISKS  115 Since C is eal, so is de (C), hence, MO o : A , GL(2,C) 1 (doo(X1)  dOo(X2)) = ( l  2) ~m2 1 De ini ion 1 . The índex o he embedding 0 0 : A -> C 2 , is he deg ee o he map A -i 80 :  --> de (M) de (M) and is deno ed by Ind(0o) E 7G . Rema k 2 . (i) I is easily e i ied ha he in ege Ind(oo) is inde- penden o he amings X and . Fo suppose ha X' and ' is ano he pai o amings, wi h ' de ined on Ll', a con ac ible neighbo hood o Oo(A) . Then he e a e smoo h maps B : un u' , GL(2,C) and C : A E -> GL + (2, R) such ha '= -B and X'=X .C . I M' : A --> GL(2, C) is he map de ined by he o mula d0 o (X) = -M', a s aigh o wa d calcula ion wi h ma ices yields he iden i y M'=B-1-M .C . l m . 2 m 2 2 de (M')  _  de (B --1 ) de (M) de (C)  _  de (B)j de (M) de (M')¡  de (B -1 ) 11 de (M) 11 de (C) 1  de (B)  Ide (M)1 Since U and U' a e con ac ible, each o he amings and ' a e is homo opic o he aming ( a a l , a a 2 ) . The map B -1 is, he e o e, homo- opic o he iden i y . This ac , oge he wi h he obse a ion ha he 3 By a eal aming we mean a pai o eal ec o ields which a e poin wise indepen- den o e R . 11 6  T . DUCHAMP, F . FORSTNERIC deg ee depends only on he homo opy class o he map, comple es he a gumen . (ii) No e also ha he abo e a gumen shows ha i q> : U , (C 2 is a biholomo phism on o an open se in (C 2 hen Ind(-P o 0 0 ) = Ind(0o) . (iii) Finally, because he index depends only on he homo opy class o he map, 00, i is de e mined by he image 00 (A) oge he wi h an o i- en a ion . Thus, i A C C 2 is a o ally eal, o ien ed, embedded annulus, he in ege Ind(A) is well-de ined . Lemma 4 . I A C C 2 is a o ally eal embedded annulus which is con ained in a o ally eal embedded disk DC C2 hen Ind(A) = 0 . P oo : Le 0 : 0 --> (C 2 a smoo h map such ha O(0) = D and choose amings X o Tá and o (C 2 . Then le M(S), E 0, be he GL(2, C)- alued ma ix as de ined abo e . The deg ee o he map ( -> de (M«))/I de (M(~))1,  E  -I (A) is ze o because i is homo opy o a cons an . 3 . Reduc ion o he case o eal analy ic bounda y Begin by assuming ha he e a e smoo h embeddings 0 and 0 o 0 wi h 0 holomo phic on 0 and o ally eal and such ha he condi ions z/~(0) 1 o(0) = D and O(á0) _ O(á0) a e bo h sa is ied . Wi hou loss o gene ali y we may assume ha 0 ex ends holomo phi- cally o a neighbo hood o 0 . To see his we obse e ha , because he condi ion o o al eali y is an open condi ion, any C l -small de o ma ion o he map 0 is also a o ally eal embedding . In pa icula , le and in such a manne ha 0o6 = {~ : 10 =1- 0, whe e S > 0 is a small cons an o be chosen la e . Then 0 can be de o med o a map 0' so ha 0'(ao) = 0(aos) o (o) n~b (oó) = 0 . TOTALLY REAL AND HOLOMORPHIC DISKS  117 Oneway o accomplish such a de o ma ion is o le 0' be he composi ion 0 o ó o  wi h he low, , o a ec o ield which is angen o he image o  and cons uc ed so ha ó (0(OO) =MO5), b > 0 . The map ~b' : 0 --> CZ de ined by he equa ion 00= )((1- b)0 ex ends holomo phically o a neighbo hood o 0 C C . Now eplace he pai 0, 0 by he pai 0', 0' . By cons uc ion, O(OO) is eal analy ic . whe e 4 . Holomo phic disks a e ela i ely iso opic o o ally eal disks Re um now o he p oblem o eplacing 0 by a o ally eal embedding . Because 0 ex ends o a holomo phic embedding o a neighbo hood o 0, he e is a biholomo phism (D : U -j C 2 , de ined on a neighbo hood U o ~b (A) such ha he composi ion <D o 0 is he map Conside he amily o maps 0 E :0_(C2, ~ H (z~ w) = (S, E«)), c .(~) =,E ( 1 - 1(I2) ex(1-I(I2) ~ No e ha E sa is ies he condi ions : o = 0, ,(~) = 0 o I(I = 1 . By i ue o equa ion (1) and he compu a ion O E(~) = E {(1 _ 21(1 2 ) - ( 1 - I(I 2 )I(I 2 i} e'(' _1C12) 0 0 he embedding de ined by 0 .is o ally eal o all E > 0 . Because o E su icien ly small he image o 0 . lies in he se <D(u), he map 0E =P -1 o 0E is well-de ined . Lemma 5 . Fo e su icien ly small, he amily 0E has he ollowing p ope ies : ( 1 ) O,Iao = Ojao o all E . (ii) 0E is a o ally eal embedding o 0 o E > 0 . (üi) z/~ E (0) CU . (i )  E(A) n o(o) _ 11 8  T . DUCHAMP, F . FORSTNERIC P oo . P ope ies (i) h ough (iii) a e immedia e om he de ini ion o , The e i ica ion o (i ) is based on he implici unc ion heo em . Fi s no e ha since O( 0 ) is o ally eal and V)0(0) is holomo phic, Tp(oo( 0 )) n Tp(o(0)) = Tp(o(a 0 )) o all p E 0(á 0 ) . Because ~b E de- pends smoo hly on e, he condi ion Tp(,pE( 0 )) n Tp(O(0)) = Tp(O(a0)) holds o all e su icien ly small . By he implici unc ion heo em and compac ness o 80, i ollows ha he e is a numbe S > 0 such ha 0E(A) U O(A6) _ 0 o all su icien ly small c . Mo oeo e , since 0o(A) n ~(0) = 0 and PE depends smoo hly on e, i ollows om he compac ness o 0(0 Ab ) ha 0, (A) n O(0 A6) _ 0 o all su icien ly small e . Hence, o e su icien ly small, 0, (A) n 0(0) = 0 . 5 . Modi ying wo o ally eal disks o abu smoo hly Conside he small annula neighbo hood Ab C 0, 0 < S < 1, o he bounda y 80 . We will modi y 0(0) on O(A6) so ha O( 0 ) and 0F( 0 ) abu in a C l manne along <b(á0) and hus de ine a o ally eal embedding o he 2-sphe e in o (C 2 . By i ue o he equali y O(ó0) = 0( 80 ), o S su icien ly small he imago O(A, )is con ained in he neighbo hood U o he p e ious sec ion . Fo his eason he map 4) o0 : A6__+ C 2 is well-de ined and, since we will modi y 0 only along A6 , he de o ma- ion o 0 can be ein e p e ed as a de o ma ion o 0' . The modi ica ion will be done in wo s ages : (i) we i s de o m 0' so ha O'(A, )is he g aph o a unc ion g ; (ii) hen we de o m g so ha O(0) and 0( 0 ) abu smoo hly . 5 .1 . Replacing O'(A . ) by he g aph o a unc ion . Conside he neighbo hoods o O'(0) o he o m N E , Q = {(z, w)  :  Iz1 < 1 + e, IWI < Q} wi h e > 0 and a > 0 chosen so small ha he e is an inclusion N3E,, C -D(U) and such ha he condi ion ~ _1 (N3e,a)n0( 0 A6) = is sa is ied . By choosing S' < S su icien ly small we can insu e ha inclusion 0'(A, - ,) CV,, is sa is ied . TOTALLY REAL AND HOLOMORPIIIC DISKS  119 We a e going o eplace 0' by ano he o ally eal embedding ~" A 6 -~ NzE,Q which sa is ies he wo condi ions : and "(Ab ) 1  {(z, g (z)) : z E Aó,}  o o,' < su icien ly small, whe e g is a complex- alued unc ion de ined on Aá, . By cons uc ion, he map 4> -1 o A b _ (C 2 ag ees wi h 0 on he in e io bounda y componen o he annulus and so de ines ano he o ally eal embedding, 01 : 0 --> (C 2 which in e sec s O(0) along he ci cle O(a0) . To begin he cons uc ion o 0" obse e ha he image O'(A6 ) is o he o m whe e he unc ions Z and W sa is y he condi ions (4)  Z(~) = ( and W«) = 0 o 1(1 = 1 . Hence, in pola coo dina es ( = pe e ", z = e e we can w i e wi h R(1, a) = 1 and O(1, a) = a . Applying condi ion (i ) o Lemma 1 yields he inequali y 0 :~ aZ aZ áp 8a aw aw ap aa on he annula egio l Aó Aó, z = Z(0,  w = W(O ,  1 - S < j(j< 1, Z= R(p, a)e2o(P,a) aZ aRe io áw ap aa 0 Thus áP is non-ze o o (p, a) = (1, a) .  Con inui y implies ha o S' > 0 su icien ly small OP does no anish anywhe e on he annulus Ab . This and equa ion (4) show ha , a e possibly dec easing S' s ill u he , he map ( H (e z© , W) is an embedding o A,, ; and, he e o e, ha o any smoo h unc ion R(p, a) > 0 he map 0' , : ( - (Z (0, w (S)) = (Re 20 , w) _ -i eZa áPW when p = 1 . de ines an embedding o A, - , . O cou se, we wish o choose R so ha he map 0" sa is ies he condi ions s a ed abo e . Tha we can do so is implied by he ollowing lemma . 12 6  T . DUCHAMP, F . FORSTNERIC Choose u" > 0 so ha m/M >C . We claim ha he e is a cons an Q"' < " and a unc ion h such ha h( ) = 0 o 1 < < 1 + o,"' , h( ) = 1 o > 1+o,"/2, and h'( ) < H( ) whe e H( ) = (Mm)-1 (( /M -C) . Tha such a unc ion exis s is clea because he igh hand side o he las inequali y is posi i e o 1 < < 1 + u" and because he in eg al l+," H( ) d di e ges o +oo . Hence we may se h( ) = i k( )d whe e k( ) is any unc ion sa is ying he condi ions, (i) 0 <_ k( ) < H( ), (ii) i +al k( )d = 1, and (iii) k( ) = 0 o 1 <_ < o,"' < u"/2 and o >o,"/2 . 1 .  E . BISHOP, Di e en iable mani olds in complex Euclidean space, Duke Ma h . J . 32 (1965), 1-21 . 2 .  F . FORSTNERIC, Analy ic disks wi h bounda ies in a maximal eal submani old o (C 2 , Ann . Ins . Fou ie 37 (1987), 1-44 . 3 .  F . W . KAMBER AND PH . TONDEUR, Cha ac e is ic in a ian s o olia ed bundles, Manusc ip a Ma hema ica 11 (1974), 51-89 . 4 .  E . L . STOUT AND W . ZAME, To ally eal imbeddings and he uni- e sal co e ing spaces o domains o holomo phy : some examples, Manusc ip a Ma hema ica 50 (1985), 29-48 . 5 .  R . O . WEI,I,S, Compac eal submani olds o a complex mani old wi h nondegene a e holomo phic angen bundles, Ma h . Ann . 179 (1969),123-129 . 6 . M . GROMOV, "Pa ial di e en ial ela ions," E gebnisse 3 olge Bd .9, Sp inge -Ve lag, Be lin-Heidelbe g-New Yo k, 1986 . Tom Duchamp : Depa men o Ma hema ics Uni e si y o Washing on GN-50 Sea Ie, WA 98195 U .S .A . Re e ences F~anc Fo s ne ic : Depa men o Ma hema ics Uni e si y o Wisconsin Madison, WI 53706 U .S .A . P ime a e sió ebuda el 12 de Ma i de 1992, da e a e sió ebuda el 10 de Juny de 1992