ANALYTIC DISCS WITH BOUNDARIES IN A GENERATING CR-MANIFOLD By Miran Cerne A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Mathematics at the UNIVERSITY OF WISCONSIN - MADISON 1994 Abstract The problem of perturbing an analytic disc with boundary in a CR-submanifold of Cn is considered. A theorem by Globevnik on the perturbation by analytic discs along maximal real submanifolds of Cn is generalized and used in various applications: (i) it is proved that every energy functional minimizing disc in C" with free boundary in a Lagrangian submanifold of C" and all partial indices greater or equal to —1 is holomorphic, (ii) a new proof and a generalization of a result by Pang on the Kobayashi extremal discs is given, (iii) perturbations of analytic varieties with boundaries in a totally real torus in C2 fibered over the unit circle dD are considered. Also, some results by Baouendi, Rothschild and Trepreau on the family of analytic discs attached to a CR-submanifold of Cn of a positive CR-dimension are globalized. Acknowledgements I am grateful to my thesis advisor Professor Franc Forstneric for his numerous inspiring suggestions and constant encouragement. I am indebted to Professor Josip Globevnik for very stimulating discussions, to Professor Yong-Geun Oh for his seminar talk and stimulating discussions, and to Professor Jean-Pierre Rosay for his lecture on the work by Baouendi, Rothschild and Trepreau. I would also like to thank my friend Manuel Flores for many math discussions we had. Last but not least, I wish to thank my wife Simona and my daughter Katja for their patience, help and constant support, and my parents Mitja and Marija for their help and support. Contents Abstract Acknowledgements 1. Introduction 2. Notation and terminology 3. Maximal real bundles over the circle 4. Some computations 5. Perturbation by analytic discs 6. Analytic varieties over the disc 7. Minimal discs with free boundaries 8. Stationary discs 9. Examples and counterexamples 10. CR-vector bundles Bibliography IV 1 1. Introduction Given an analytic disc in Cn with boundary in a generating CR-submanifold MCC", one would like to describe the family of all nearby analytic discs in C" attached to M. This problem is the object of a considerable research in the recent years. The following list of authors and their papers related to the problem is not at all meant to be complete : Alexander [Ale2], Baouendi, Rothschild and Trepreau [Bao-Rot-Tre], Bedford [Bedl, Bed2], Bedford and Gaveau [Bed-Gav], Eliashberg [Eli], Forstneric [For2, For3], Globevnik [Glol, Glo2], Gromov [Gro], Y-G.Oh [Ohi, Oh2], Tumanov [Turn]. The technique of perturbing an analytic disc with boundary in a given manifold has found several applications in the problems of the analysis of several complex variables. Two, probably the most known problems, where this technique can be used, are the problem of describing the polynomial hull of a given set in Cn and the problem of extending CR-functions from a given CR-submanifold of C" into some open subset of C". Recently has J.Globevnik in his paper [Glol], which was inspired by the work [For2] by F.Forstneric, found very elegant sufficient conditions on a given analytic disc p with boundary in a maximal real submanifold M of C" which imply finite dimensional parametrization of all nearby holomorphic discs attached to M. To each, not necessary holomorphic, disc p with boundary in a maximal real submanifold MCC" one associates n integers ki,... ,kn called the partial 2 indices of the disc p. Their sum k := k\ + • • • + kn is called the total index of the disc p. A part of Globevnik's work [Glol] is the theorem in which he proves that if the pull-back bundle p*(TM) of the tangent bundle T M is trivial and if all partial indices of the disc p are greater or equal to 0, then there exists an n + k dimensional parametrization of all nearby discs of the form p + analytic disc with boundary in M. Later we proved that Globevnik's theorem extends in the same form to the case where the pull-back bundle p*(TM) is non-trivial. The final version of the result was given by Y-G.Oh in [Ohi], where the Globevnik's result is generalized, but using a different approach, to the case where all partial indices are greater or equal to —1 and arbitrary pull-back bundle p*(TM). This theorem, together with the papers [For2, For3] by Forstneric, represents the starting point of the present thesis and is reproved in its most general known form, using only Forstneric's and Globevnik's technique, in section 5, Theorem 1. The present work is organized as follows. Section 2 introduces the notation and terminology we use throughout the work. In section 3 the maximal real bundle over the unit circle dD C C and its partial indices are defined, and in section 4 some computations of the partial indices of a maximal real bundle over dD are given. As already mentioned, in section 5 we reprove the generalized version of Globevnik's theorem using his and Forstneric's technique of perturbing 3 analytic discs with maximal real boundary conditions. In the following sections we give some applications of Theorem 1. In section 6 we apply Theorem 1 to the problem of perturbing analytic varieties with boundaries in a totally real torus in C2 fibered over the unit circle dD C C. In section 7 we consider energy functional minimizing discs in C" with Lagrangian boundary conditions and prove that the condition that all partial indices of the disc p are greater or equal to —1 implies that the disc p is in fact holomorphic. Section 8 considers stationary discs which are, following Slodkowski [Slo4], related to the problem of describing the polynomial hull of a fibration over the unit circle dD with the fibers in C" and to the problem of finding Kobayashi extremal discs through a given point in an open set in Cn. Using Theorem 1 again, we reprove and generalize a result by Pang, [Pani], on the Kobayashi extremal discs. In the next section we give several examples which show that the immediate generalization of the continuity method to describe the polynomial hull of a set fibered over the unit circle with fibers in Cn, n > 2, as used by Forstneric, [For3], in the case of one dimensional fibers, is even in some relatively simple cases impossible. In the last section we first extend Globevnik's results to the case of analytic disc attached to a generating CR-submanifold of C" and then also generalize some results by Baouendi, Rothschild and Trepreau, [Bao-Rot-Tre], to large analytic discs with boundaries in a generating CR-submanifold of Cn. 2. Notation and terminology Let D = {z G C; I z I < 1} and let dD denote the unit circle in C, the boundary of D. If K is either D or 3D, and 0 < a < 1, we denote by C°'a(K) the Banach algebra of Holder continuous complex-valued functions on K with finite Lipschitz norm of exponent a imi i ,i . \f(x) - f(y)\ / ll/Ha = SUp l/l + SUp ---- _ ---- < 00 • XÇlK x,yeK \X y\ For every mGNU {0} we also define the algebra Cm,a{K) = {fe cm. y^ = ^ ^f^ < Qo} _ \j\ £ e c^ sucn that 0(O = 0o(r(O) (£ed£\{i}). In other words, this is the space of continuous functions g on dD\{l} such that a) there exist the limits (1) lim g(eie) and lim g(eie) which we denote by g(l+) and g(l~), respectively, and are related by the equation g(l+) + g(l-) = 0 , b) the function (2) {H9m'-={g-î(e) ; Im£ > 0 ; Im| < 0 is in C™,a(dD). Obviously £™,a is an R-linear space and for the norm on it we take \\n\\ .— WfJnW in C Pm'a\ \\y\\m,a ¦— ll-^i/llm,« \y t O-^ j . So £™,a is a Banach space that is via H isometrically isomorphic to the closed subspace of odd functions in C™'a(dD). Remark. Another equivalent description of the space £™,a can be given in terms of Fourier series. Namely, each element g G £™,a has a unique expension of the form oo k=—oo where the sum oo ^ Cke" 0i(2k+l) k=—oo represents the Fourier series of some odd function g0 G C™,a(dD). We will refer to Cjt, k G Z, as the Fourier coefficients of the function g. One can define a Hilbert transform T on £™,a. Let T0 be the standard harmonic conjugate function operator on C™,a(dD). Then is defined by Tg = H-lT0Hg (g G £^a) . Note that T0 takes the subspace of odd functions in C™,a(dD) into itself. Thus for every g G £™,a the function H(g + iTg) := Hg + *#T# = Eg + iT0(^) is an odd function on dD from the space Am'a(dD). We denote the space of functions of the form g + tTg (g G £^a) 7 by Am'a. Observe that all functions from the space Am,a are of the form r (£)/(£) for some / G Am,a(dD). Observe also that since for any two functions g and h from £™,a the following identity holds (Hg)(t)(Hh)(Q = g(e)h(e) (£ G dD), the product of two functions from £™,a gives a function in C™'a(dD), and the product of two functions of the form g + iHg, g G £™,a, gives a function in Am'a(dD). The spaces we will most often consider are the finite products of the spaces C^a(dD) and £R'a. A product with n factors will be denoted by £a, where a is an n-vector with O's and l's as its entries. The entry 0 on the j-th place represents the space C^a(dD) as the j-th factor and the entry 1 on the j-th place means that the j-th factor is the space £ffi'a. By analogy we also define the spaces Aa which are the products of finitely many copies of A0,a(dD) and A0,a We extend the definition of the Hilbert transform in a natural way (componentwise) to the space £a. We denote the extension by Ta. It is a bounded linear map from £a into itself and it has the property that the vector function v + iTav belongs to the space Aa for every v G £a. We also define the map Ha:£a^ (C^(dD)) 8 which is defined as the identity map on each factor CR'a(<9D), and is defined as the map (2) on each factor £R'a. 3. Maximal real bundles over the circle Let L be a maximal real subspace of Cn, i.e., its real dimension is n and L®iL = Cn. To any such maximal real subspace L one can associate an R-linear map Rl on C", called the reflection about L, given by z = x + ix i—> x — ix (x, x G L) , where z = x + ix is the unique decomposition of z into the sum of vectors from L and iL. The mapping RL : C" —? Cn is an R-linear automorphism of Cn which is also C-antilinear, i.e., Rl(ìv) = —iRL(y) for every v E Cn. The reflection about the maximal real subspace Rn C C" will be denoted by R0. Note that in the standard notation R0 is just the ordinary conjugation on C" and that for any n x n complex matrix A the following identity holds A = R0AR0 . LEMMA 1. Let L be a maximal real subspace of Cn and let Xi,... ,xn be any set of vectors spanning L. Let A := [xi,... ,xn] be the matrix whose columns are the given vectors Xj, j = 1,..., n, and let B := AA~l. Then B = RlRq • 9 Moreover, the matrix B does not depend on the basis of L, i.e., B remains the same even if a different basis for L is selected, and B = B~l , \detB\ = 1 . Remark. In the above lemma n x n matrices A and B are identified with C-linear automorphisms of Cn in the standard basis. Proof. Observe that A is a C-linear automorphism of Cn which maps Rn onto L. Consider the following composition of automorphisms of C" S := R0A~lRLA . Then S is a C-linear automorphism of Cn which equals to the identity on Rn. Since Rn is a maximal real subspace of Cn, S is the identity on C", and hence R^-1 = A-XRL . Finally, since B = AlF1 = ARoA^Ro , we get B = AA~lRLR0 = RLR0 . The rest is obvious. H The following definition is taken from [Glol], see also [For2]. 10 DEFINITION 1. Let L = {LÇ;Ç G 3D} be a real rank n subbundle of the product bundle 3D x C" of class C0,a. If for each £ G 3D the fiber L^ is a maximal real subspace of Cn, the bundle L is called maximal real. Example. A very important example of a maximal real bundle over 3D one gets in the following case. Let M be C2 maximal real submanifold of Cn and let p : 3D —>¦ M be a C2 closed curve in M. Then the pull-back bundle p*(TM), where T M is the tangent bundle of the submanifold M, is a maximal real bundle over 3D of rank n. It is known, see [Vekl], that for every closed path B in Gl(n,C) of class (jo,a one can ßn(j holomorphic matrix functions F+ :D^Gl(n,C) , F~ ¦ C\ D —? Gl{n, C) of class C°'a and n integers ki > k2 > ¦ ¦ ¦ > kn such that b = f+(0H0f-(0 (te 3D), where j Ìkl 0 ...... 0 \ 0 ffc2 0 0 a(0:= . . _ . . ;;; .. • \ 0 ...... 0 Çfc" / The matrix A will be called the characteristic matrix of the path B. One can prove that under the condition k\ > ¦ ¦ ¦ > kn, the characteristic matrix A does not depend on the factorization of the matrix function B of the above form, see 11 [Veki], [Gioì], [Cla-Goh] for more details. The integers k\,..., kn are called the partial indices of the path B, and their sum K := K\ -|- • • • + Kn is called the total index of the matrix function B. DEFINITION 2. Let L be a maximal real bundle over the unit circle 3D. The partial indices of the Gl(n, C) closed path (3) BL : £ >—> Äl.Äo o/ c/ass C°'a; are ca//ed t/ie partial indices of the bundle L and their sum is called the total index of L. Remarks. 1. Observe that Definition 2 makes sense even if the bundle L is not trivial. 2. The total index of a closed path p on a maximal real submanifold MCC™ is also called the Maslov index of p. 3. As we will see, in the case where all partial indices satisfy the condition kj > — 1, j = 1,..., n, the characteristic matrix A(£) carries all important information about the bundle L, see also [Gioì], [Ohi]. Although Globevnik in [Glol] works only with the trivial bundles over the circle 3D, Lemma 5.1 in [Glol] still applies and one can conclude 12 LEMMA 2. The C°>a closed path in G7(n,C) Bl-.Z^R^Ro (ÇedD) can be decomposed in the form BL(0 = e(£)A(Oë(|F (£ G ÖD) , where the map © : D —> Gl(n, C) zs o/ class C0,a and holomorphic on D, i.e., the n x n matrix © is in A°'a(D)nxn. Let the total index k be an even integer. Then one can split the matrix A as where Q(Ç) is a closed real analytic path in Gl(n,C). See [Glol] for details. Fix £ G <9D and select any basis xi,..., xn of the fiber Lç. Let A := [xi,..., xn] be the matrix whose columns are the vectors Xj's. Then bl(0 = af1 = e(e)Q(ç)(6(ç)Q(e))-i and so ^_1e(OQ(0 = A-ie(ç)Q(ç). Thus the invertible n x n matrix u := A-^COQCO is real and therefore the columns of the matrix 0(£)Q(£) = AU span the fiber Lç for each £ G dD. Together with a Globevnik's observation, see also [For2], one concludes 13 COROLLARY 1. A maximal real bundle L over 3D is trivial if and only if its total index is an even integer. According to [Bot-Tu], every real vector bundle over 3D of rank n is either trivial or isomorphic to the direct sum of a trivial bundle of rank n — 1 and the Möbius bundle. Since the trivial bundle case was discussed in details in [Glol], one would only have to consider the non-trivial bundle case. But since our approach to the problem does not "see" the difference between the trivial bundle case and the non-trivial bundle case, we will still consider both cases. Let L be a rank n maximal real C0,a vector bundle over 3D. Let &i > &2 > ¦ ¦ ¦ > kn be its partial indices and let / £fcl 0 ...... 0 \ 0 fk'2 0 0 a(0:= . _ . . ;;; .. • \ 0 ...... 0 Çfc" J As we already know the C0,a closed path in Gl(n, C) BL:Ç^RLiR0 (Ce 3D) can be decomposed in the form bl(o = 0(ç)A(ç)ë(ëp (e e dD) for some O : D —> Gl(n,C) of class C0,a and holomorphic on D. The characteristic matrix A can be decomposed further as A = A^ = A0X^ , 14 where f& MO := 0 0 \ 0 o ç- 0 .. V o ...... o i^ ) k Here £2 stands for £m if k = 2m and for £mr(£) if /c = 2m + 1. We will refer to A0 as the square root of the characteristic matrix A and we say that the matrix function i ^ e(£)Ao(0 (e e 9D) represents the normal form of the bundle L . To the C0,a closed path BL in Gl(n, C) we also associate the corresponding Danach space £a we will work with, see section 2 for the definition. The n-vector a is defined as a := (ki mod 2,..., kn mod 2) . COROLLARY 2. // a// partial indices of a maximal real bundle L are non-negative, then there exists annxn matrix function A(Ç), £ G 3D, with the rows from the space Aa and such that its columns Xi (£),..., Xn(£) span the fiber Lç for every £ G <9D. Remark. For £ = 1 the above statement still makes sense in terms of the limits (1) when £ ^ 1 approaches to 1. 15 4. Some computations Let (•, •} denote the standard inner product on C". Let L be a maximal real subspace of C". By iL we denote the maximal real subspace of vectors of the form iv, v G L, and by LL the maximal real subspace of vectors perpendicular to L, i.e., a vector u G Cn is perpendicular to L if and only if Re(n, v) = 0 for every v G L. We recall that RL denotes the C-antilinear reflection about the maximal real subspace L and that the matrix BL is given as the product RlR0- LEMMA 3. a) Let L be a maximal real subspace of C". Then RiL = —Rl -, BiL = —BL and i?Lx = -WL = Rl , ßLx = -Bi . b) Let L = {Lg; £ G 3D} be a maximal real bundle over the circle 3D. Then the following holds : 1. The partial indices of the bundles L and iL are the same. 2. The bundles L, iL and LL are trivial if and only if one of them is trivial. 16 Proof. Let AL, AiL and AL± denote matrices whose columns span the subspaces L,iL and L-1, respectively. Then AiL = %AL and Re(A*LAL±) = 0 . Hence BiL = MlA~l = —Bl and BL, = A^ÄZ = -JÄ^A\AL,A-L\ = -Bi . Part (a) is proved. Part (b) is now a trivial consequence of part (a) and Corollary 1. ¦ Remark and an example. One should notice that the indices of the normal bundle L1- are not always the same as the indices of L, e.g., if the matrix function BLis \ 0 ÌJ ' then its partial indices are 1 and — 1, but on the other hand the partial indices of B\ are all 0. Of course, one also has to check that BL = B^1. M LEMMA 4. Let L0 be any (trivial or non-trivial) maximal real bundle of rank n over the circle 3D and let A0(£), £ G 3D, be a C0,a path in GL(n,C) which represents the normal form of the bundle L0. 17 1. Let L be a maximal real bundle of rank n + 1 whose fibers are spanned by the columns of the matrix ' 9 0 A - ' v A0 where g is a nonzero function from the space £k0moà2 with the winding number |, fc0 G Z, and v is any vector function from {£k0moà2)n¦ If ki,... ,kn are the partial indices of the bundle L0 and if kj — k0 > — 1, j = 1,..., n, then the partial indices of the bundle L are "»oj "'lj • • • j ™n • 2. Let L be a maximal real bundle of rank n + 1 whose fibers are spanned by the columns of the matrix it, 0 A^ - « e«(o mo where v is a vector function from the space (A°'a(dD))n. If ki,... ,kn are the partial indices of the bundle L0, then the partial indices of the bundle L are Z, K\, . . . , Kri Proof. 1. Let B0 = AoA-1. Then Br = AA- 9/9 0 (v-B0v)/g B0 18 Once we find a solution (a, b) G (A°'a(dD))n+1 of the equation (4) BK)(g)=^(«(|)) Ke3ß) such that the function a extends as a nonzero holomorphic function on D the first part of the lemma will be proved. Since the winding number W(g) is 4f, the function g can be written in the form g(0=p(mko/2em (tedD), where p is a positive function of class C0,a and h belongs to the space A0,a(dD). Let a := eh. Then the first equation in (4) is solved and the second equation has the form (5) =(w - B0v)ê + B0b = ikob . Let $A$_1 be the normal splitting of the path B0 and let Multiplying (5) by <3>_1 from the left-hand side yields hv-AV)e1> + Aß = Ck°ß . 9 Thus for each j = 1,..., n we have the equation kko/2(vj-e^)+e3Wj = e°ßj- 19 After dividing by £fej/2 and rearranging the terms, the problem we are trying to solve is to find holomorphic functions ßj G A0,a(dD), j = 1,..., n, such that Im(^-M/2/^) - ~^fkjl2vM)) = 0 (É e 3D) . But this problem is equivalent to the problem of finding real functions Uj, j = 1,..., n, from the space £(k-k0)mod2 such that the function extends holomorphically to D. Since k j — k0 > — 1, j = 1,..., n, such functions is indeed possible to find, namely, let «i:=iT(^f^(0), where T is the Hilbert transform on the space £(fc--fc0)mod2-2. Let B0(£) = A0(QA0(Ç)-\ C G dD. Then -e o BL(0 = .4(0.4(0- = ^ tievi0 _-mm m) ) K € 3D) . In this form it is easy to check that the vectors *+<«>=U)) and *"K)=r2(^i) «eaB) solve the Hilbert boundary value problem and that 2 is the order of zero of the function <3>~ at the infinity. H Example. The following example shows that in the case (1) of Lemma 4 one 20 really needs some assumptions on the partial indices of the matrix function A0 and the winding number of the function g. If A is the matrix i 1 then the matrix BL = AA_1 is e o 2i£ 1 and one can easily check that 2 is not one of its partial indices. Moreover, both partial indices are 1. H The next lemma shows that a fiber preserving diffeomorphism of D x C" which is holomorphic on each fiber and biholomorphic as a mapping from DxC™ into itself does not change partial indices of a maximal real fibration over 3D. LEMMA 5. Let $ : DxCn —>DxCn $ : (e,z)^-> (£,<£(£,*)) be a C0,a(D, C1(C")) fiber preserving diffeomorphism of D x Cn such that the function (/>(£, •) is holomorphic for each £ G D and the mapping $ is a biholo-morphism of D xC. Let L be a maximal real bundle over 3D. Then the partial indices of the maximal real bundles L and L, Zç:=Dz$(ç,0)Lç (ÇEdD), are the same. 21 Proof. Let The C0,a matrix function P is holomorphic on D and, since $ is a diffeomor-phisms of D xC™, invertible on D. Also, b~l(0 = P(OBL(OP^lC) té e dD) , where BL and B^ are the corresponding Gl(n, C) closed paths (2) of the bundles L and L, respectively. Since as soon as the partial indices are ordered, the characteristic matrix A of the path B^ does not depend on the factorization of B^ of the form B~L = F+AF- , where F+:D—>Gl(n,C) and F~ : C\ D —? Gl(n, C) are holomorphic, the proof of the lemma is completed. H 5. Perturbation by analytic discs The problem we consider in this section is the following. Problem. Given a smooth map e ^ m(o (e e od) , 22 where each M(£), £ G 9D, is a maximal real submanifold of C", and a smooth map p:dD —>Cn such that p(£) G M(£) for each £ G 3D, find all smooth maps ip : D —> Cn, holomorphic on D, which are close to the zero map and satisfy the condition (p + rf(C)eM(C) (ÇedD) . This problem was also considered by Globevnik, see [Glol, Problem 1.2], for the orientable bundle case, and by Forstneric in C2, [For2]. See also the paper [Ohi] by Y.-G. Oh. The arguments we use in this section closely follow those used by Globevnik in [Glol] and Forstneric in [For2], and so not every detail will be given. The smoothness of the Problem will be C0,a for some fixed a G (0,1). That is : a) The map p : d D —>¦ C" is of class C0,a. b) For each £0 G 3D there a neighbourhood Uç0 Ç 3D of £0, there is an open ball Bço C C" centered at the origin and maps pf°,..., p|° from the space C°'a(t/fo,C2(5fo)) such that for each £ G Uîo we have 1. M(0 n (p(0 + Bîo) = {ze P(0 + B^pfit, z - p(0) = o, j = l,...,n} , 2. pf°(£,0) = 0, j = l,...,n, 3. d,p?° (£, *) A • ¦ • A ^ (£, z) ^ 0 for all 2G%. 23 An object of the above form will be called a C0,a maximal real fibration over the unit circle 3D with C2 fibers. Obviously each maximal real fibration over 3D induces a maximal real vector bundle over 3D, i.e., the bundle (J {C} x TmM(0 , iedD and hence it makes sense to talk about the partial indices and the total index of a maximal real fibration over 3D. We define them as the indices of the corresponding maximal real vector bundle. Let B : 3D —>¦ GL(n, C) be the corresponding C0,a closed path in GL(n, C) defined by the map (3) which factors as (6) B(Ç) = ^(OAfê^CI) = 4,(0 V(Ö (e e 3D) , where A0(Ç) stands for $(£)A0(£). The n x n matrix A0(Ç) has the property that for each £ G 3D its columns span the tangent space Tp^M(C) and that its rows belong to the space £a. One would also like to get a set of defining functions for the family M(£) which would reflect the splitting (6). LEMMA 6. There exist an r0 > 0 and functions pÇeC^idDiC^Bj) (l*) ! kJ isodd , mç' ' " \ (fj(Ç,z) ; fy is even , the following holds a) M(£) n (p(£) + A-J = {z G p(0 + ^ro;Pj(e^-p(0) = 0,j = 1,... ,n} , b) (9zpi A • • • A dzpn ^ O on 3D x 5ro. Proof. For each point 90 G [0, 27t] it is easy, using the definition of a maximal real fibration over 3D and some linear algebra, to find a neighbourhood Uqo Ç M of #0 and functions p\ ,..., pß from Co,a(exp(U0o), C2(Broj) such that for each £ G exp(Ue0) we have M(£) n (p(£) + 5rJ = {z G p(£) + £ro; /<(£, z - p(0) = 0,j = l,...,n} and VzPoo(e,0) = 2^(£,0) := 2(tAj1(0)* = -2t($-1)*A0 , where Pe0 = (pi , ¦ ¦ ¦ ,Pg )¦ Here exp(Ue0) denotes the open set {eîo; 0 G W^} Ç 3D. By the compactness we select a finite subcover {Wj} of the interval [0, 27r]. We may even assume, without loss of generality, that each of the points 0 and 2-7T is covered only once, and that for the sets of defining functions p0 and p27r 25 one has 4(eie,z) = fÌt(ei0,z) for every j such that the partial index kj is an even integer and PÌ(eW, z) + ÛM9, z) = 0 for every j such that k j is an odd integer. Let {xj} be a smooth partition of unity on [0, 2n] subordinated to the cover {Uj}. We define p(é\z):=YJXMPeMie^)-j For r0 > 0 small enough the above function p satisfies the required properties. Of course, the properties of the component functions pj for such subscripts j that the partial index kj is an odd integer, follow from the fact that the j-th column of the matrix A0 changes its sign when the argument arg(Ç) runs from 0 to 2n. Finally, for a subscript j such that the partial index kj is an even integer, we define the function p° to be the function pj, and for a subscript j such that kj is an odd integer we define the function p° as Hçp(-,z). Here H is the map (2) defined in section 2. ¦ Using the vector notation we define as F(u, v) := p(-, A0(u + i(v + iTav))) . 26 Observe that F is well defined and that is of class C1, see [Glol, Lemma 6.1]. Observe also that if a pair (u0, v0) G £a x £a solves the equation F(u,v) = 0 , then the boundary of the disc p + A0(u0 + i(v0 + iTŒv0)) lies in the given maximal real fibration, i.e., for each £ G 3D we have p(0 + 4,(0 MO + i(v0(Ç) + *(7>0)(Ç))) G M(0 • The rest is completely standard. First one should use the implicit mapping theorem in Banach spaces, [Car], for the mapping F and the space £a to get a parametrization of all, not necessary holomorphic, nearby discs with boundaries in the maximal real fibration {M(OiçedD- Let dp denote the matrix whose columns are the coefficients of the (0,1) forms dpi,..., dpn. For each £ G 3D we have op(£,0)Mo(0 = i/n, where In denotes the n x n identity matrix. Then the partial derivative of F with respect to v at the point (0, 0) G £a x £CT, applied to a function z/ G £CT, is in the matrix notation given by ((D5F(0,0)H(C) = 2Re(9p(É,0)M0(0(^(0 - (7>)(Ç))) = -2//(Ç) for every £ G 3D. Hence the partial derivative of the mapping F with respect to variable v is an invertible linear map from the space £a into itself. So the 27 implicit mapping theorem in the Banach spaces gives neighbourhoods V\ and V2 of 0 G £a, and a unique C1 mapping

) = 0 if and only if v = (p(u) . Finally one would like to select from the above family of all possible closed C0,a curves in the maximal real fibration {M(^)}^egD near p, those which bound a sum p + analytic disc . This is the point where one can effectively use the normal form of the maximal real bundle p*(Tp^M(C)), £ G 3D, over the circle 3D. And this is also the point where one should assume that all partial indices of p*(TM) are greater or equal to —1. For the final argument one should first observe that for the case when all partial indices of the maximal real bundle p*(TM) are greater or equal to —1, the vector function A0{v-\-iT^v) extends holomorphically to D for every v G £a. For the nonnegative partial indices this follows immediately but for the partial indices which equal to — 1 the above claim follows from the fact that for any odd partial index kj the function Vj + iTvj is of the form r(£)#?(£) for some function g? G A0,a(3D). 28 So the condition for the vector function e.—? a0(u + i(v + iTavM) (e e öd) to extend holomorphically to D is in the case kj > —1, j = 1,... ,n, equivalent to the condition that the vector function e ^ A0(ç)u(o (e g 9D) extends holomorphically to D. To detect all possible u's which have the above property one has to find all vector functions a G (A0,a(dD))n such that on dD A<2 = a , i.e., for all j = 1,..., n ^^(0 = ^(0 (ÇedD). For each partial index k j = — 1 there is only one solution of the above equation, namely, a j = 0. For k j > 0 one gets k j + 1 dimensional parameter family of solutions. A parametrization \& of all functions u G £a such that the vector function A0u extends holomorphically to D is for each component function Uj given by k ;/2 (7) tfj(*o,...,**,•)(£) :=*0 + R*É(*28-i+î*2S)O in the case the partial index kj is an even integer and by (*,-l)/2 (8) ^(t0,...,tfcj)(e):=Re( £ (t2s + *t2s+1)r(e)2s+1) s=0 29 in the case kj is an odd integer. See [Glol] for more details. Hence, altogether one has E (ki + !) = E ki + El = E ki - E1+n =k+n /y>0 fcj>0 /y>0 fcj>0 fej<0 parameter family of solutions of the Problem. THEOREM 1. Let M(£) Ç C", Ç G dD, be a C°>a maximal real fibration over the unit circle dD with C2 fibers and let p:dD —>Cn be a C0,a closed path in C" such that Pit) e M(0 (£ e ÖD) . // all partial indices of the maximal real fibration M(£), £ G <9D; a/onp ^e pa£/i p are greater or equal to —1 and the total index of p is k, then there is an open neighbourhood U of 0 G Wl+k, an open neighbourhood W of p in (C0,a(dDj)n and a map tf : U —>• (C7°'a(öD))n o/ c/ass C1 suc/i t/iat 1) tf(0)=p, 2) /or eac/i t E U Ç Wl+k the map \t(£) —p extends holomorphically to D, 3) tf(ti)^tf(t2) Zorita, 30 4) if p G W satisfies the condition p(Ç) G M(£), £ G 9D, and is such that p—p extends holomorphically to D, then there ist E U such that \t(£) = p. Remark 1. The above theorem was first proved by Forstneric in [For2] in the case where the ambient space is C2. Next, it was proved by Globevnik [Glol] in C" for the case where the pull-back bundle p*(TM) is trivial and where all partial indices are nonnegative. Later I observed that Forstneric's and Globevnik's tehnique also works in the case when the pull-back bundle is nontrivial but the indices are still nonnegative. In the meantime the Problem was, although in a little bit different context, solved by Y-G.Oh [Ohi] who noticed that the partial indices can be even taken to be greater or equal to —I. The above theorem thus also includes his observation but in the context of the Forstneric's and Globevnik's tehnique to tackle the Problem. For more on the history of the Problem, partial results and applications one should also check the papers by Alexander [Ale2], Bedford [Bedl, Bed2], Bedford and Gaveau [Bed-Gav], Eliashberg [Eli], Gromov [Gro]. Remark 2. As in [Glol, Theorem 7.1] one could also add to the above theorem perturbations of a maximal real fibration over the unit circle and the condition that in the case where all partial indices are greater or equal to 1, the set of the centers \&(i)(0) — p(0), t EU, contains an open set in Cn. 31 6. Analytic varieties over the disc Let M be a maximal real submanifold of C". The problem we consider in this section is a very special case of the following general question : Given M and V Ç C" \ M, a purely one-dimensional analytic subvariety with boundary in M, can one "find and describe" all purely one-dimensional analytic subsets in C" \ M that are near V ? Let n = 2 and let n : C2 —>¦ C be the projection on the first coordinate plane n : (£, z) >->¦ £. Let T Ç dD x C be a compact, connected, totally real two-dimensional submanifold of C2 of class C2 such that for each £ G 9D the fiber Tt = {zeC;(Ç,z)eT} is the union of q, q G N, simple closed curves in C whose polynomial hulls are pairwise disjoint. Observe that T is a totally real embedded torus in C2. In our setting M C C2 will be a finite disjoint union of totally real tori fibered over the unit circle 3D CCx {0} Ç C2, i.e., k M=\jT] i=i where 71,12, ¦ ¦ ¦, Tk are pairwise disjoint totally real tori in dD x C Thus each fiber Mç = {zGC;(Ç,z) e M} 32 is a finite pairwise disjoint union of q = q\ + • • • + qu Jordan curves J|,..., Jh For V, a purely one-dimensional analytic subset of C2 \ M, we assume that is a ç-sheeted analytic variety over D, i.e., a) V Ç D x C is given by a Weierstrass polynomial of degree q V = {(Ç,z)eDxC;zcl + cuiOz«-1 + • • • + aq(Ç) = 0} , where 0,1,0,2, ¦¦¦ ,aq are in the disc algebra A(D), b) V n Jl ^ 0 for every j = 1,2,... ,k and every £ G 9D. Observe that by a theorem of Cirka, [Cir], o\,..., a,q are also in A2~°(D). Given M and V as above one can construct a compact connected maximal real manifold M in Cq+1 and an analytic disc V with boundary in M in the following way : Each fiber M n 7r_1(£), £ G 9D, is the disjoint union of g Jordan curves J\, j = 1,..., q. Let 7f := Jj x • ¦ • x J| CO. We define the map $ : C9 —? Cq (zl,...,zq) 1—>¦ {sl,...,sq) where 1 " sp = ^I]^ (P = 1, ••-,?) • ^ .7=1 33 Observe that $ is a proper map from Cq into itself and that its Jacobian determinant is Mz) = det(D$(z)) = \{{z3 - zt) . By the assumptions on the manifold M, i.e., the curves JL j = 1,... ,q, are pairwise disjoint, the g-dimensional tori 7ç, £ £ dD, do not intersect the branch locus of the map $, i.e., the set of points in Cq where J$(z) = 0. Moreover, by the same reason and since the coefficients of a polynomial uniquely, up to the order, determine its zeros, $ is injective on each 7ç. The image of 7ç under <3> is denoted by 7ç and is a maximal real ç-torus in Cq of class C2. We define M:= Utf}x7i. iedD Since locally, over some arc I Ç 3D, M is given as the image of Uçe/{£} x 7ç under the map C x C —> C x C (t,z) —? & <&(*)) and since the component functions of the maps <3>ç are symmetric in their arguments, M is a maximal real compact connected manifold in Cq+1 fibered over d D of class C2. By an assumption on V the points in the fiber V Pi 7r_1(£) represent exactly one point in 7ç. The mapping <3> maps each of them into 7ç. Since V is given 34 by a Weierstrass polynomial and since <3>ç is given by elementary symmetric functions of its arguments, one concludes that the boundary of V is mapped into the boundary of an analytic disc in Cq+1 attached to M. Moreover, this disc is given as the graph of an analytic disc in Cq which takes £ G d D into 7ç. One also observes that the varieties over D with boundaries in M which satisfy the above conditions are in one to one correspondence with the graphs of the analytic discs in Cq which map £ G d D into 7ç. Namely, a basis of symmetric functions in q arguments Zi,...,zq can be given either with the functions Si,..., sq or with the symmetric functions which one gets through the Vieta formulae. So the problem of finding and describing all analytic varieties with boundaries in M near to the given analytic variety V with boundary in M is now translated to the problem of finding all analytic discs with boundaries in M close to the given one. Remark and an example. Although the fibers 7ç of the manifold M are maximal real ç-dimensional tori, the manifold M itself is not necessary a q + 1-dimensional torus as the following example shows : Let the manifold M be given by M:={(t,z)(=dDxC;\z2-Ç\ = ±} 35 and the variety V by the equation z2 — £ = |. Then the pull-back of the tangent bundle of the 3-dimensional totally real manifold M C C3 along the graph of the associated analytic disc is not trivial. Thus M is not a 3-torus. H We denote by p : 3D —> Cq the corresponding analytic disc whose graph has boundary in M and by p' its derivative. Fix £0 G 3D. Since the curves JL j = 1,..., q, Ç £ 3D, are pairwise disjoint, one can, locally near £0, order the roots of the Weierstrass polynomial defining the variety V. Let Q!i(£) G J|,..., cxq(Ç) G Jf denote its roots. Also, to each root %(£), j = 1, • • •, q, Ç G <9D, there corresponds a unique unit outer normal t,-(£) to the curve J| Ç C at the point ct/(£). Using local parametrization of M, definition of the manifold M, and the above notation, a matrix A whose columns span T^^^M can be written as A 0 ^(0 ìc*(0t(C) where «(?) ttl(Ç) «2(C) 3-1/ and r(0 v «r(o 0 r2(0 lì Ì 0 \ V o 0 r,(0 / Thus if one defines 5(£) := a(Ç)r(Ç)(a(Ç)r(Ç))-1 and £(£) := AA~\ £ G ÖD, then the proof of Lemma 4 part 2 implies that one of the indices of the 36 Gl(q + 1, C) closed path B equals 2 and the remaining q indices are given by the Gl(q, C) closed path B. COROLLARY 3. The total index k0 of the closed path B is given by g k0 = 2W([[rJ) + W([[(a,-atf) j=i j>t where W(g) denotes the winding number of a function g G C (dD) with no zeros. Proof. The total index k0 is also given as the winding number of the determinant det(.B). Using the special form of the matrix B and the fact that the matrix a is the Van der Mond matrix the corollary follows immediately. One should also recall that the winding number of the product of two functions equals to the sum of the winding numbers of its factors. H Results from the previous section and [Glol] now imply PROPOSITION 1. // all partial indices of the Gl(q, C) closed path B are a) (Existence) greater or equal to — 1, then near V there is a k0 + q + 3 parameter family of analytic varieties with boundaries in M. Moreover, if each partial index of B is at least 1, then the family of analytic varieties with boundaries in M that are near V contains an open subset of C2. b) (Nonexistence) negative, then there is a neighbourhood of V in C2 such that in this neighbourhood the variety V is the only analytic variety with boundary in M. 37 COROLLARY 4. A necessary condition that all partial indices of the Gl(q, C) closed path B are positive, and thus the union of the family of analytic varieties with boundaries in M close to the variety V contains an open subset of C2, is 2W(f[r]) + W(ll(a]-aty)>q. i=i j>t Proof. A necessary condition that a family of nearby varieties with boundaries in M contains an open set of C2 is that all partial indices are greater or equal to 1. So the total index k0 has to be greater or equal to q. M Example. Let the manifold M be given by M:={(^z)edDx£-\z2-e\ = \} and the variety V by the equation z2 — £2 = |. The normals to the fibers of M at the boundary of V are given by ^¦(0=^(|j (J = l,2) . Therefore W(tj) = —1, j = 1,2, and W(a2 — cki) = 1, and our necessary condition to have a lot of nearby analytic varieties with the boundary in M fails. One can also calculate the partial indices for this example. They are 2, 0 and -2. ¦ 7. Minimal discs with free boundaries Let M C C" be a C2 manifold and let p : D ->• C" be a disc of class C1^ with boundary in M, i.e., p : 3D —>¦ M. By the energy of the disc p we mean 38 the Dirichlet integral of p E(p) := ö / \Vp\2dxdy . 2 J D The maps which minimize the energy functional are of a special interest in the Riemannian geometry, namely, any map with boundary in a submanifold M which minimizes energy E in a certain homotopy class [p(dD)] éGÇ m(M), also minimizes the area functional A(p) :-- dp a dp dx ' * dy dxdy ' D in G, see [Nit] and [Ye2] for more details. The advantage of the energy functional E with respect to the area functional A is that the first is only conformai invariant but the latter is invariant under any diffeomorphic change of coordinates. Let (r, 9) denote the polar coordinates on the unit disc D. It is well known, see e.g., [Jag], [Lew], [Ye2], that any solution p of the Euler-Lagrange equations for the energy functional E with free boundary in the manifold M must satisfy the following conditions : a) All component functions of the mapping p are harmonic in D, and b) ^(Oi-Tp(0M for every £Ì«»-o 40 and so A short calculation shows ime2um7(ëy> = o for every £ G 3D. But the function extends holomorphically to D and so F(£) is a constant function. Since F(0) = 0, one concludes that n 3=1 for every £ G 9D. COROLLARY 5. Let p be a complex function p : D —>¦ C o/ ^e c/ass C1'". // either of the discs F, G : D —>• C2 , w;/iere F(*) = (*,p(*)) , G(z) = (ž,p(z)) , is minimal for some submanifold M in C2, then the function p is either holo-morphic or antiholomorphic. In particular, M can be a totally real torus in C2 fibered over the unit circle 3D. 41 Proof. Let us assume that the disc F(z) = (z,p(z)) is minimal for some submanifold M. Then the corresponding holomorphic discs / and g are of the form f(z) = (z,a(z)) and g(z) = (0, b(z)) for some holomorphic functions a and b from A1,a(D). The above argument then implies a'(z)b'(z) = 0 for every z G D. Thus at least one of the functions a and b has to be a constant function. In the case the disc G(z) = (z,p(z)) is minimal, the proof is similar. ¦ We would like to use the special normal form of the pull-back bundle to investigate the energy functional minimizers with boundaries in a Lagrangian submanifold M Ç Cn of class C2. PROPOSITION 2. Let L be a maximal real n-dimensional vector bundle over 3D of class C0,a such that every fiber Lç, £ G dD, is a Lagrangian subspace of Cn, and let BL be the C0,a closed path in Gl(n,C) which represents L. Then : a) B*L = B^1 so the matrix BL(Ç) is unitary for every £ G dD. b) BlL = BL, and conversely, if for a maximal real bundle L over dD one has B\ = BL, then the bundle L is Lagrangian, i.e., each fiber is a Lagrangian subspace of Cn. 42 c) Let n0 denote the number of partial indices of the bundle L which are greater or equal to —1. Then there are n x n0 and n x (n — n0) dimensional matrix functions X and Y, respectively, with entries from the space A°'a(dD) such that the columns of the matrix function [X,Y]A0 span the fiber L^ for each £ G 3D. Moreover, XlY = 0 on 3D. Here A0 is the square root of the corresponding characteristic matrix A. Remark. In the case n0 is n (resp. 0) the matrix Y (resp. X) does not exist. Proof. Since the fibers of the bundle L are Lagrangian subspaces of C", the bundles iL and L1- are the same. Lemma 3 implies BfL = BL and then also, by Lemma 1, B*L = B^1. This proves (a) and the first part of (b). To prove the reverse implication of part (b) one should observe that if A is any n x n matrix whose columns span Lç for some £ G 9D, then AÄ^ = BL(£) = Bl(Ç) = Ä^A* = (A*)-1 A1 . So A* A = AfA = A^J and the matrix A*A is a real n x n matrix. Hence Re((iA)*A) = 0 43 and so the columns of the matrix %A are perpendicular to the columns of the matrix A. Part (b) is proved. To prove part (c) we use the factorization of the matrix function BL, BL = $A• Gl(n,C) from A°'a(D)nxn. We define \t := ($-1)* and by part (b) we also have BL = ^"vF1 . Let $ = [$l5 $2] and \I> = [^i, ^>2] be the block notation of the matrix functions <3> and \t such that the matrices $1 and \&i have dimensions n x n0. Since $^ = $*($-1)t = In , we get and so $*^ = 0 . This means that the columns of the matrix <3>! are orthogonal to the columns of the matrix ^2, and so the matrix [$i,*2] is invertible. We set X := $x and Y := ^2 and the proof is finished. H &2 44 The proof of part (c) of the above proposition also implies COROLLARY 6. Let L be a Lagrangian vector bundle over dD of class C0,a, let BL be the corresponding C0,a closed path in Gl(n, C) and let A be its characteristic matrix. Then the characteristic matrix of the Gl(n,C) closed path is A. Remark. Statement (c) of the above proposition does not hold for an arbitrary maximal real vector bundle over the 3D. As a counterexample one can take the bundle whose matrix B is given by r?f) It is easy to check that the partial indices of B are 2 and —2 and that there are no nontrivial holomorphic functions a,b G A0,a(dD) which satisfy the equation o f )\m J-e \W) Henceforth M will denote a C2 Lagrangian submanifold in Cn and p : D —> Cn will be an energy functional stationary disc of class C1,a with boundary in M, i.e., the first variation of the energy functional E at p is 0. We recall that 45 the pair (r, 9) represents the polar coordinates on D. Then one has and for every £ G 9D. Since the disc p is attached to the manifold M, the vector o§(Š) is tangent to M at p(£) for each £ G 9D. Since p is minimal, the vector dp dr #(£) is perpendicular to M at p(£) for each £ G <9D. Let A0 denote the matrix constructed in part (c) of Proposition 2. Since M is a Lagrangian submanifold of C" we get Re(^*i(ç)) = 0' and since the first variation of the energy functional E at p vanishes, we have Re(^f(e)) = 0. Thus for each £ G ÖD we get Re(Um^(0) = 0 and Re(^(£)||(0) = 0 • Since p = f + ~g for some vectors /, g from Al,a(dD)n, we have (9) MÇAto(Ç)g'(0) = 0 and Re(^(C)/'(£)) = 0 . Since also A0 = [X, Y]A0 where the matrices X and Y have holomorphic extensions into D and are "orthogonal" to each other, i.e., YtX = 0, we first conclude 46 that X*g' = 0 and Y* f = 0 , and finally /' = Xh and g' = Yk for some vector functions h G (^4°'a(D))n° and k G (A°>a(D))n~n°. For these results we used only one part of each equation in (9). The rest, together with the above equalities, implies Re((X+X*Xh) = 0 and Re((X^Y*Yk) = 0 , where A+ and A~ are the n0 x n0 and the (n — n0) x (n — n0) dimensional matrices, respectively, such that A = ( K -) Since Re(-iA*0A0) = 0 on 3D, the matrices Ä+X*XA+ and Ä^Y*YA~ are real and invertible. Thus R*(£Â+(ë)MO) = 0 and Re(^Ä^)k^)) = 0 for every £ G 3D. This proves the following theorem. 47 THEOREM 2. Let M be a C2 Lagrangian submanifold in C" and let p = f + g, f,g E (Al>a(ydD))n, be the energy functional stationary disc of class C1,a with boundary in M. Let n0 denote the number of partial indices of the path p(dD) Ç M which are greater or equal to —1, and let X and Y be the nxn0 and n x (n — n0) dimensional holomorphic matrices given by part (c) of Proposition 2. Then there exist vector functions h G {A°>a(dD))n° and k G (A°>a(dD))n~n° such that Re(eX+(l)MO) = ° and MtK¥)HO) = o (e e dD) , and f = Xh and g' = Yk . Remark. In the case n0 equals n (resp. 0) one part of the above conlusion is empty, namely, the matrix Y (resp. X) does not exist. As a simple consequence one has the following COROLLARY 7. (Hypothesis as above.) 1) // all partial indices of the pull-back bundle p*(TM) are greater or equal to —1, then p is a holomorphic disc. 2) // all partial indices of the pull-back bundle p*(TM) are less or equal to 1, then p is an antiholomorphic disc. 48 8. Stationary discs Let 9Qç, £ G 3D, be a C1'" family of strongly pseudoconvex C4 hypersur-faces in Cn, i.e., there exists a C^a{dD, C4(Cn)) function p = p(£, z) on 9Dx C" such that for every £ G 9D a) the hypersurface dfìc equals to the set {z E Cn; p(£, z) = 0, dzp(£, z) ^ 0} and b) the domain {z E Cn; p(£, z) < 0} is strongly pseudoconvex. Let f : D —> Cn be a holomorphic map of class C1,a such that fio g 9Qç (e g öd) . We will call such / a holomorphic disc in C" with boundary in the family of strongly pseudoconvex hypersurfaces {dQ^}^egn- For any such mapping / we define M0:=f^,/(0)- The following definition seems to be a natural extension of the definition in [Lem], see also [Slo4] and [Pani]. 49 DEFINITION 3. The disc f is said to be stationary if and only if there exists a C1,a positive function p on dD such that the mapping e—?Ptéwco (e g öd) extends as a holomorphic mapping on D with no zeros. Remark. In the cases considered by Lempert in [Lem], by Slodkowski in [Slo4], and by Pang in [Pani] the geometry of the problem under consideration implies that if p is a positive C0,a function on dD such that the mapping £ i-» P(OM£) extends holomorphically to D, the extension is nonzero on D. We recall the Webster's construction in [Web] where he showed that the natural embedding of a C2 hypersurface ECC™ into Cn x CPn_1 via the map (10) $>:z^(z,T^E) (zeE) is maximal real near a point (z0,TzcE) if and only if the Levi form of E at z0 is nondegenerate. Thus the image of a strongly pseudoconvex hypersurface E under this natural embedding is always a maximal real submanifold of C" x CP™"1. Using the natural duality, i.e., the space of complex hyperplanes in C" is naturally biholomorphic to the space of complex lines in C", observe that every stationary disc / with boundary in the family of strongly pseudoconvex hypersurfaces {0Q^}^egD, induces an analytic disc e-->(/(0,[M0]) 50 in Cn x CP"-1 attached to the maximal real fibration {^(9Qc)}ceaB. So it makes sense to talk about the partial indices of a stationary map, i.e., we define the partial indices of a stationary map as the partial indices of the induced map. Notice that to each stationary map / we associate 2n — 1 partial indices. LEMMA 7. Let h : D —> C" be a holomorphic disc of class Ck, k > 1, such that h(0 ^ 0 (e G D) . Then there exist holomorphic discs h2, h3,..., hn of class Ck, k > 1, such that det(h(0,h2(0,---,hn(0) = l (£€£). In particular, the vectors h(Ç), h2(Ç),..., hn(£) are linearly independent for every Remark. The lemma was inspired by Proposition 9 in [Lem]. Proof. We will prove the lemma by induction on the dimension n. For n = 1 the claim is trivial. For n = 2 the lemma follows from the fact that since the component functions / and g of the mapping h, i.e., h = (/, g), have no common zeros and since the space Ak(D) of the k times differentiable holomorphic functions on D is a Banach algebra with a unit where the holomorphic polynomials are dense, then the characterization of the maximal closed ideals in Ak(D) as in the proof of Theorem 18.18 in [Rudi] implies that there exist holomorphic 51 functions F and G of class Ck such that /(0^(0+ 0(0^(0 = 1 (ieD). Then the mapping />2(0 := (-G(0,n0) (£eä) is such that det(/i(0, ^(0) = 1 f°r every £, £ D- Let us assume the lemma for n > 2 and we will prove it for n + 1. Let /i : D —>¦ C"+1 be a holomorphic disc of class Cfc with no zeros on D. After a linear change of coordinates in C" one may assume that the first n component functions of the mapping h have no common zeros on D. This follows since h can also be considered as a Ck mapping from D into CPn and therefore it can not be surjective. Let h = (g, hn+i) where g stands for the first n components of the mapping h. By the inductive assumption one can find g2,..., gn analytic mappings from D into C" of class Ck on D such that det(g,g2,...,gn) = 1 on D. Now the mappings hj-.= (9j,0) (J = 2,...,n) and /in+1 := (0,..., 0,1) prove the lemma for the dimension n + 1. H 52 Henceforth let f : D —> Cn be a stationary map for the family of strongly pseudoconvex hypersurfaces dQç, Ç G dD, in C". Therefore there exists a positive function p on <9D of class C1,a such that £>(£)/i/(0> £ e ^-^j can be extended as a nonzero holo-morphic mapping on the disc D. The above lemma for k = 1 implies that the function p is essentially unique, i.e., if pi and p2 are two real functions on dD such that Pißf and P2ßf extend as holomorphic mappings on D with no zeros, then there exists a positive constant a such that Observe that here we really need an assumption on the zeros of the maps Pi^f, i = 1,2, namely, e.g., the function £^Re(£ + 2)Ç (C e dD) extends as a holomorhic function on D but the real function p(£) = Re(£ + 2) is not constant. We normalize the map £>(£)/i/(0> £ e dZ^> so that at the point £ = 0 the length of the extended holomorphic vector is 1. The so obtained map we call, following Pang [Pani] (although our definition differs a little bit from his), the 53 dual map of /, and we denote it by /. Since / has no zeros on D, one can use the above lemma to get a C1 holomorphic frame /, h2,..., hn over D. We define two fiber preserving diffeomorphisms of D x C" which are holomorphic on each fiber and biholomorphic as mappings from DxC" into itself. Such a diffeomorphism does not change partial indices of a C0,a closed path in a family of totally real submanifolds, Lemma 5. The first one is <&i:(e^)^(ç,z-/(e)) and the inverse of the second one is n *? ¦¦ (t,*) —? (£ ,/(0*i + E wo**) ¦ The composition $ := $20$i is then a C1 fiber preserving diffeomorphism such that in the new coordinates, i.e., after applying $, the stationary disc / and its dual / have extremely simple form, namely, f(t) = 0 and f(0 = (1,0,..., 0) (ÇGdD). We still denote the defining function of the family of strongly pseudoconvex hypersurfaces 9Qç, £ G 3D, by p = p(£, z) and we may assume, without loss of generality, that gf"(Ç, 0) = 1 for every £ G 3D. Since the hypersurfaces dflç are strongly pseudoconvex for each £ G 9D, the complex Hessian of the function p(£> ") is positive definite when restricted to the maximal complex tangent space of 9Qç at the point 0, T0cdfìc = {z G C"; zx = 0} = C™"1 , 54 i.e., the (1,1) minor L0(£) of the complex Hessian of p(£, •) represents the Levi form of the hypersurface dQ^ at 0. Thus the (n — 1) x (n — 1) matrix function of class C°'a on dD i ^ Lo{i) satisfies the theorem [Lem, Théorème B] and so there exists an (n — 1) x (n — 1) matrix function K, K :D—>GL(n-1,C) , of the same smoothness and such that a) K is holomorphic on D, b) K*(Ç)K(0 = Lo(0 (tCdD). After using another fiber preserving diffeomorphism on D x Cn, where z' stands for (z2, • • •, zn), one may assume that /(0 = 0 > /(O = (l,0,...,0), L0(Ç)=Id (ÇGdD). We will now compute the total index of a stationary map. Later on we will apply the same kind of computation to find all partial indices of a stationary disc under some geometric assumptions on the family dQç, £ G dD. 55 We denote by In the n x n identity matrix and by Jn the (n — 1) x n matrix ( ° * 0 0 0 1 0 \ 0 ^ 0 0 0 1J The other matrices we need are t«»:=(a*4K'0))iJ.1 ; H(0 _ / 92p \dzidzj and H0:~- - JniiJn . (e,o) «j=i Since the derivative pZl (£, 0) equals 1 for each £ G <9D, the mapping ^ : Cn - C2n-1 P*2 (£, *) P^n (£, *) - P*l(£,2)''"' P*l(£,z) is well defined in a neighbourhood of dD x {0}. Notice that ^ç restricted to dil^ is just the Webster's map (10) written in the local coordinates. A short computation shows a'*<<°>=U*) ¦ ^(o)=(A) ¦ Since the columns of the matrix I l\^\ 5 O j. j L>J\ span the tangent space to dVt^ at the point 0 and since for every v G C" we have Dz*t(0)v = dz*t(0)v + dz*t(0)i 56 the columns of the matrix l€\ Jn %Jn i(JnHei — JnLei) JnHJn + JnLJn i(JnHJn — JnLJn) span the tangent space to ^(ôfiç) at the point (0,0). Using our notation we can simplify the above matrix as (¦\-i\ ( ïel '-'n %'-'n [ ' {iJnHer Ho + I^ i(H0 -1^) We compute the determinant of the above matrix by expending along the first row. We get that the determinant of (11) equals to ¦In—1 ^-*n—1 H0 + In-i i(H0 — In-i) We multiply by i the first column and subtract it from the second to get 4-1 o H0 + In-\ — 2iln-i Thus the determinant of the matrix (11) equals to îdet(-2îL0) =idet(-2îln_l) = (-2)™"^" and we proved the following proposition. PROPOSITION 3. The total index of a stationary disc f with boundary in a family of strongly pseudoconvex domains in C" is 0. To compute the partial indices of this path in the family of maximal real sub-manifolds {^(dQç)}çeQD one first has to find the inverse of the (2n — 2) x (2n —2) 57 matrix The inverse is A ±n—1 î-1-n—1 1 Ho + In-1 i(H0 — In-l) A i x / 4-1 ~~ H0 I, o n—1 2 I -i(H0 + 4_i) î/„_i and so the matrix 50 := A0A0 x is [ —H0 ___ In-i V 4-1 ~~ H0H0 H0 We recall Definitions 3.7 and 3.9 from [Pani] which in our context are DEFINITION 4. The family of hypersurfaces 9fiç, £ G 9D, is strongly convex along f(dD) if and only if the real quadratic form on C"_1 v i—> \v\2 + Ke(H0v ¦ v) is strongly positive definite. We also say that the family 3Qç, £ G 3D, is strongly convexifiable along a stationary disc f if there exists a fiber preserving biholomorphism of DxCn such that in the new coordinates the family {3Ç}ç}çegD is strongly convex along f(3D). Remark. Observe that the condition on a family of hypersurfaces to be strongly convex along f(3D) is slightly weaker than the strict geometric convexity of hypersurfaces 3Qç at /(£) for each £. For example, let the hypersurface 9Qç for each £ G 3D be defined by the equation p(z1,z2) := 2Re(zi) + \z2\2 + 2Re(zf) . 58 Then, at the point (0,0), L0 = 1 and H0 = 0 for every £ G 9D and so this family of hypersurfaces is strongly convex along the disc /(£) = 0, £ G 9D. On the other hand p{t + Vl*| + 2t2, 0) = 2(t - |t|) - 2t2 < 0 for any real number t Gl Hence the domain fiç lies on both sides of the hyperplane Re(zi) = 0 and can not be convex. PROPOSITION 4. // dQç, £ G 3D, is a family of strongly pseudoconvex hypersurfaces in C" which is strongly convexifiable along a stationary disc f, then all partial indices of f are 0. COROLLARY 8. If all hypersurfaces 3Qç, Ce 3D, are strictly geometrically convex, i.e., the real Hessians of their defining functions are positive definite, then all partial indices of any stationary disc of this fibration are equal to 0. Remark. This was the situation studied by Slodkowski in [Slo4] and Lempert in [Lem]. Proof.(Proposition) Since the sum of the indices of a stationary map is 0, it is enough to prove that there are no positive partial indices. Let us assume that there exists a positive partial index k0 of the path B0. By the definition of partial indices there exist holomorphic discs a and b in Cn_1, with no common 59 zeros on D, of class C°'a, and such that on 3D B-(«(l)=fbU!>) Keöß)- So for every £ G 9D We conjugate the above identity and dot it by a to get -H0(Ç)a(Ç) ¦ a(0 + 6(C) • a(£) = f °W) • «(6 • Multiplication by £feo and taking the real parts of the equation yield the following pointwise equation on 3D (12) Re(e*"a(0 ¦ 6(0) = l«|2 + Re(#0(ÇW2a(e)) ¦ (£*°/2a(0)) . Since the family of hypersurfaces 9Qç, £ G 9D, is strongly convex along f(dD), the right hand side of (12) is positive for every fixed £ G ÖD. On the other hand the function Re(£feoa(£) • 6(C)), Ç G D, is harmonic on D and equals 0 at the point 0. The mean value property for harmonic functions gives a contradiction to the assumption that there exists a positive partial index of the path B0. Lemma 4 completes the proof of the proposition. H Remark. One can easily observe that the proof of the above proposition also works in the case where the fibers of the fibration dQç, £ G 3D, are only convex for each £ G 3D and strongly convex on a subset with a positive Lebesgue measure. 60 One can observe that in the above proof only the upper part of the matrix B0 was used to derive a contradiction. Next lemma tells us that this was not just a coincidence. LEMMA 8. Let a and b be two C0,a holomorphic discs in C"_1 such that (13) -H0(Ç)a(Ç) +W)= e°a(t) (Ç G 3D) . Then the pair (a, b) solves the equation B-(«(l)=fbU«) Keafl)' Proof. We recall that B = ( ~H° __ In-X \ V In-i — H0H0 H0 J We would like to show that the discs a and b also solve the equation (14) (/„_! - H0($lÜäMS + H0(OW) = ÇkoKÇ) (e e dD) . We rewrite (14) as H0(m-Ho(o — l\t d* / o J-n—1 tl0tl0 \B0 S = B0 = \ T -JT- \ -»ri-I -no where we also used the property that Ht0 = H0. Thus if a pair (a, b) of two C0,a holomorphic discs solves the equation B"K) ( w> ) = e"{ m ) «e dD) for some k0 G Z, then the pair (—io, io) solves the equation Hence the partial indices of the matrices B0 and B* are the same and we proved the following PROPOSITION 5. Ifk0 is a partial index of the Gl(2n-2, C) path B0, then —k0 is also a partial index of the path B0. We recall the definition of a non-degenerate stationary disc from [Pani] but in a modified form. See [Pani] for more details. 62 DEFINITION 5. A stationary disc f is said to be non-degenerate if the equation has only the trivial solution in the space (A0,a(dD))n~1, i.e., a pair of vector-functions ß and 7 from the space (A0,a(dD))n~1 solves the above equation if and only if ß = 7 = 0. PROPOSITION 6. The only possible partial indices of a non-degenerate stationary disc are 0,1 and — 1. Remark. Note that by Theorem 1 and by an observation by Slodkowski, [Slo4], this proposition immediately implies Theorem 4.8 from [Pani]. Proof. Let (a, b) be a nontrivial pair of functions from (^4°'a(<9D))n_1 which solves the problem B°{0 ( 1 ) = e" ( m ) «e «» —2 for some k0 G N. Then, after the multiplication by £ , the first n — 1 equations can be rewritten as We consider two cases. 63 1. k0 = 2m + 2 for some integer m > 0. The multiplication by £ yields the equation ë^W) = (r «(O) + e#o(e)(e"a(e)) and the non-degeneracy of the stationary disc implies a = b = 0 , a contradiction. 2. k0 = 2m + 3 for some m > 0. Then the multiplication by £ gives Let /9(C) := £ma(£) and 7(C) := £,m+1b(£). Then the above equation has the form After the multiplication by £ we also have er(Ü = /9(0 + WMWW) • Adding both identities yields m+imo = (e+i)/(o+e#o(e)((e+1)/(0) • Thus, by the non-degeneracy of the stationary disc, the functions /3 and 7 are identically 0 and so also a = b = 0 , a contradiction. Lemma 4 finishes the proof of the proposition. H 64 9. Examples and counterexamples Example 1. The first example shows that there exist a real analytic family of real analytic hypersurfaces in 3D x C" with strongly pseudoconvex fibers and a family of corresponding stationary analytic discs such that the partial indices associated to each stationary disc change for some isolated values of the parameter. Let p|(*i,z2) = 2Re(z!) + \zx\2 + \z2\2 - tRe(lN z%) ((t,£) eRxdD) be a two parameter family of strongly plurisubharmonic functions on C2. For each pair (t, () G 8x 3D the function p| defines in a neighbourhood [7| of the point (0, 0) a strongly pseudoconvex hypersurface E| given by the equation p\{z) = 0. If one restricts the parameter t G M on a compact subset /ÇR, the neighbourhoods [7|, t G I,Ç G dD, can be chosen uniformly. It is clear that p|(0,0) = 0 and 9zp*(0,0) = (1,0) . Let Ml denote the maximal real submanifold of the complex manifold C2 x CP1 which one gets as the image of the strongly pseudoconvex hypersurface E| by 65 the mapping * : z _> (Zi [dp\(z)]) . See the section on the stationary discs and [Web]. Claim. The partial indices of the closed curve (15) £ ^ (0,0, [1,0]) (ÇedD) are a) all 0 for t ^ ±1, b) 0, N and -N for t = ±1. Proof. A computation in the coordinate chart V = {(z, [w]) G C2 x CP1; W\ ^ 0}, similar to the one in the proof of the fact that the total index of a stationary disc is always 0, gives a matrix At(£) whose columns span the tangent space to the maximal real submanifold M^ at the point (0, 0, [1, 0]), i.e., at the point (0, 0, 0) in the coordinates, i 0 0 MO := [ o i o ] (ç e dD) . —N , —N . o -te +i -%{f +i) So we get -10 0 Bt(t) = At(t)At(t)-i = | 0 tÇ» 1_ | (È 0, the constant has to be 0, and thus 6 = 0 . Going back to the equation (17) one gets (l-t2Më) = 0 (ÇGdD). Thus if t ^ ±1, the function a has to be identically 0, which gives a contradiction to the assumption k0 > 0 and so in the case t ^ ±1 all partial indices of the 67 curve (15) are 0. For the case t = 1 one can check that one of the partial indices is TV, namely, the pair of holomorphic functions (1,0) solves the equation (16) for k = N. The pair of functions (i, 0) shows the same for t = —1. Of course, the second partial index is — N. M Example 2. Examples 2 and 3 together with Example 1 will show that the so called continuity method for describing the polynomial hull of a general hypersurface in 3D xC", n > 1, with strongly pseudoconvex fibers fails. This is in contrast with the case n = 1, where the continuity method was successfully used by Forstneric, [For3], to describe the polynomial hull of a totally real torus fibered over 3D. We will find an isotopy of hypersurfaces Et, t G [0,1], in 3D x C2 with strongly pseudoconvex fibers which starts at a hypersurface E0 in 3D x C2 whose fibers are Euclidean spheres in C2, is strictly decreasing in the sense that Et is included in the domain bounded by the hypersurface ET for all r, t < t, and ends with a hypersurface Ei in 3D x C" with the property that its polynomial hull is nontrivial but there is no graph of a bounded analytic disc with boundary almost everywhere in the hypersurface Ei. See also Example 3. Let 7 be a smooth arc in R2 C C2 and let / be any smooth nonnegative function on R2 such that a) the zero set of / and the zero set of the gradient V/ are both equal to 7 and b) there exists an r0 > 0 such that f(xi,x2) = x\ + x\ for x\ + x\ > r2. 68 Here the coordinates in R2 C C2 are xi,x2 and the coordinates in C2 are Z\ = X\ + iyi and z2 = x2 + iy2. For A > 0 we define px(zi,z2) = f(xi,x2) + Kvl + yl) ¦ Then a) the zero set of p\ and the zero set of Vp\ are both equal to the arc 7 and b) the Levi form of the function p\ is T(n \ ._ _ I Jxixi ~r cA JxiX2 j 4 y JxiX2 JX2X2 ~r ^ / where the notation fXiXj stands for the second partial derivative of the function / with respect to Xi and Xj, i,j = 1, 2. So if A is large enough, the function p\ is strictly plurisubharmonic on C2. We fix such a A and denote the function p\ by p. Let x : R ^ [0,1] be a smooth function whose support is contained in the interval [—1, (7*0 + 2)2] and which equals 1 on the interval [0, (r0 + l)2]. Also, let g be a smooth nonnegative function on E. such that 1) g(x) = 0 for x < r2, 2) g'(x) > 0 and g"(x) > 0 for x > r2, 3) p(z)x'(\z\2) +g'(\z\2) > 0 for every z G C2. For e G (0,1) we define 69 If e is small enough, the function p is strictly plurisubharmonic on C2 and its zero set is the arc 7. We fix such an e and denote the corresponding function by p. Claim. The zero set of the gradient Vp is the arc 7. Proof. Let z° = (x° + iy°,x% + iy%) be a point where the gradient Vp is zero. We consider the following three cases : 1. \z°\ < r0. Then p = ep in a neighbourhood of the point z° and thus z° G 7. 2. \z°\ > r0 + 2. Then p(z) = p(|2:|2) in a neighbourhood of the point z°. Since g'(x) > 0 for x > r2, we get a contradiction. 3. r0 < \z°\ < r0 + 2. The y components of the gradient Vp, i.e., the derivatives of p with respect to y\ and y e(p(z)X'(\z\2) + 0 on C2, it follows 2/1=2/2 = 0. This, together with the fact that \z°\ > r0 and our initial assumption (b) on the function /, implies Jxi\xiix2l = ^Xi and jX2\Xi,x2J = ^x2 • 70 The x components, i.e., the derivatives with respect to x\ and %2 variables, of the equation \7p(z°) = 0, together with (18) give x° = x°2 = 0 . Thus also the assumption r0 < \z°\ < r0 + 2 leads to a contradiction and the claim is proved. H. Thus for every simple arc 7 in R2 C C2 we found a smooth parameter family of strictly pseudoconvex hypersurfaces Et, t G [0,1], in C2 which starts at 7, it is strictly increasing in the sense that for each pair of parameters t < r the hypersurface Et is included in the interior of the domain bounded by ET and which ends at some large Euclidean sphere. Remark 1. If one is given a smooth family of simple arcs 7c, £ G dD, in R2 C C2, then one can choose a smooth family of smooth functions /ç, £ G dD, satisfying the conditions (a) and (b) for each £ G dD. Since the set of parameters is compact, the functions x and 9 and the constants A and e can be chosen uniformly, i.e., independent of the parameter £ G dD. Remark 2. The above construction can be applied to any arc 7 in C2 for which there exists an automorphism $ of C2 such that $(7) Ç R2. 71 We consider the following family of arcs in R2 C C2. Let 71 be the semicircle in R2 given by the equation Xl + X2 = 1 ) X2 > 0 . For each £ G dD we denote by Rç the map Rç : C2 —? C2 defined by i?ç(zi,£2) := (^1,^2) • Observe that R% is a linear isomorphism of C2. For £ G OD such that 0 < arg(Ç) < f or ^ < arg(Ç) < 2tt let 7ç := 7i • For the parameters £ G 9D such that | < arg(Ç) < ^ we smoothly perturb the initial arc 71 to get arcs 7c such that they do not pass through the point (0,1) but they still pass through the points (1,0) and (—1,0). For instance, for £ = els one may take 7c to be defined by the equation {l-Q{s)fx\+xl = {l-Q{s)f , X2>0, where g : R —>• [0,1) is any smooth function whose support is the interval [|, ^]. We define 7ç := Retiti ¦ 72 Here by -*/£ we mean the principal branch of the square root, i.e., vT = 1. Since we have 7^ = 71 in a neighbourhood of £ = 1 and since the arc 71 is symmetric with respect to the a^-axis, the family of arcs 7c, £ G 3D, is smooth. Using our initial construction for one arc 7 C R2 and Remarks 1 and 2, one gets a smooth family of hypersurfaces Et, t > 0, in 3D x C2 such that for each t all their fibers are strongly pseudoconvex and for t large enough all the fibers of the hypersurface Et are equal to a sphere of a fixed radius \fi centered at the point (0,0). Also, for every pair t,r Gl+, t < r, the hypersurface Et is included in the domain bounded by ET. Remark. Observe that by a theorem of Docquier and Grauert [Doc-Gra] the above properties of the isotopy Et, t > 0, assure that the closures of the fibers of the domain bounded by Et remain polynomially convex for each time t. To finish our example we first observe that since (Vë,0),(-Vë,0)e7ç (te 3D) , the polynomial hull of Et contains the point (0, 0, 0) for all t > 0. Finally we prove the following claim. Claim. For t > 0 small enough there is no graph of a bounded analytic mapping F : D —>¦ C2 whose boundary is almost everywhere with respect to the Lebesgue measure on 3D contained in the closure of the domain bounded by 73 the hypersurface Et Ç 3D x C2. Proof. We first prove the claim for So := (J {£} x 7Ç • iedD Once this is proved the normal family argument and the above remark finish the proof of the claim. Let us assume that there is an analytic mapping (f,g):D^e such that (/(£)>0(O) e 7f (a.e. ÇG9D). Therefore the imaginary part of the function g is almost everywhere 0 on 3D and thus g is a constant function, i.e., there is a real number a G [0,1] such that g(Ç) = a for every £ G 3D. Since the arcs 7ç for | < arg(Ç) < ^ do not pass through the point (0,1) the constant a has to be less than 1. But then /(02 = (1 - al)i (a.e. Ç G 9D) , which leads to a contradiction. ¦ Example 3. In this example we will construct a smooth family Et, t G [0,1], of smooth hypersurfaces in 3D x C2 similar to the one in the Example 2, i.e., E0 = 3D x S2n~1(R) for some R > 0, the family is strictly decreasing in the sense that Et is included in the domain bounded by ET, r < t, and all the 74 fibers Et n ({£} x C"), £ G 9D, are strongly pseudoconvex for each value of the parameter t, but this time we will also have a fixed neighbourhood dD x B(e0) of 9-D x {(0,1)} included in every domain bounded by Et, t G [0,1], and there will be a point in the polynomial hull of Ei that can not be reached by the graphs of bounded analytic discs with boundaries almost everywhere in E^ Let 7 Ç R2 C C2 be the arc as in the Example 2. Let Xx := 7 and let ^ç := ^V?^1 • Since again (Ve,0),(-Ve,0)GXc (ÇG9D), it is obvious that the polynomial hull of X := (J U} X XÇ cedo contains the point (0, 0, 0). Claim. There is no graph of a bounded analytic disc F : D —>¦ C2 whose boundary is almost everywhere with respect to the Lebesgue measure on dD contained in X and which passes through the point (0, 0, 0). Proof. Assume that there is an analytic disc F = (/, g) whose graph has 75 boundary almost everywhere contained in X and such that F(0) = (0, 0). This implies, similarly as in the Example 2, that 9(0 = 0 (ÇGD). Thus /2(Ç) = e (a.e. i e 3D), a contradiction. ¦ The rest is similar to the Example 2 and thus omitted. Remark. Examples 2 and 3 were inspired by the example by Helton and Merino in [Hel-Mer] where they constructed a connected and simply connected fibration over the unit circle 3D with a nontrivial polynomial hull and such that there exists no graph of an analytic disc with boundary in the fibration. 10. CR-vector bundles We begin with a definition. DEFINITION 6. Let L = {Lç Ç Cm+";Ç G 3D} be a real vector bundle over 3D of class C0,a. If for each £ G 3D the fiber Lç is a real vector subspace of CR-dimension m, the bundle L is called a CR-bundle of CR-dimension m over the unit circle 3D. If, in addition, for each £ G 3D the fiber L^ is a generating subspace of Cm+n, i.e., Lç + iL^ = Cn+m, £ G 3D, then the bundle L is called a generating CR-bundle over 3D. 76 Remarks. 1. Every maximal real bundle over dD is a generating CR-bundle with CR-dimension 0. 2. To each CR-bundle L over the unit circle one can associate a complex m-dimensional vector subbundle Lc C L which is just the bundle of the maximal complex subspaces of the bundle L, i.e., for each £ G dD the fiber L^ equals to Lç n iLç. LEMMA 9. Let V be a C1 complex vector bundle over 3D such that for each £ G dD the fiber Vç is an m dimensional complex subspace ofCm+n. Then there exists a linear change of coordinates in Cm+n such that in the new coordinates each fiber Vç projects isomorphically onto Cm x {0} Ç Cm+n. Moreover, the set of invertible (m + n) x (m + n) matrices satisfying this property is open and dense in Gl(m + n, C). Proof. We denote by Ç the set of invertible (m + n) x (m + n) complex matrices having the above property. Clearly Q is open in Gl(m + n, C). So, to prove the lemma, we have to show that the complement of Ç in Gl(m + n, C) has no interior. Fix £0 G dD. Let A^o be any (m + n) xm matrix such that its columns form a basis of the fiber Vço. We define the mapping $Ço :Gl(m + n,C) —> C 77 by $to(U) = det([Im,0]UAto) where [Im, 0] is an m x (m + n) matrix which has the identity matrix in its first m columns and the 0 matrix in its last n columns. The mapping <3>ço depends on the matrix A^o, i.e., on the basis of the fiber Vç0, but the set does not. The equation is algebraic and so U^0 is an algebraic subset of Gl(m + n, C). Hence, locally the set Uç0 has finite 2((m + n)2 — 1) dimensional Hausdorff measure. Let U := (J {£} xUçÇdDx Gl(m + n, C) . Then, locally again, the 2(m + n)2 — 1 dimensional Hausdorff measure of the set U is finite and so for every compact set K Ç Gl(m + n, C) we have %2{m+nY-li^{U) DK) <00 where n is the projection ¦K-.dDx Gl(m + n, C) —> Gl(m + n,C) ¦ Since tt(U) is exactly the complement of the set Ç, the lemma is proved. H 78 Let E C Cm+" be a generating CR-subspace of CR-dimension m such that its maximal complex subspace Ec projects isomorphically onto Cm x {0}. LEMMA 10. The subspace S:=En({0}xCn) is a maximal real subspace of {0} x Cn and is the only subspace of {0} x C" for which Proof. We denote by n : Cm+n —>• Cm x {0} the orthogonal projection onto Cm x {0}. Since n projects Ec isomorphically onto Cm x {0}, we conclude that for every ïGE there exists exactly one vector «eEc such that ir(x) = ir(v). Hence the vector x — v G E is in the kernel of the projection n, i.e., x — v is in {0} x C". Therefore x — v is in S. The assumption on the projection n also implies (19) Ecn({0}xCn) = {0} , and so S is a totally real subspace of {0} x C" for which E = Ec © S . Finally, the subspace E is a generating CR-subspace of Cm+" and thus S is a maximal real subspace of {0} x Cn. The uniqueness follows from (19). H 79 Let L C 3D x Cm+" be a generating CR-bundle of the class C0,a over the unit circle and of CR-dimension m. We assume that each fiber L^, Ç G 3D, projects isomorphically onto Cm x {0} Ç Cm+n. By Lemma 9 this assumption can always be realized in the case where the bundle L is of class C1. The above Lemma 10 implies that there is a unique maximal real bundle C Ç 3D x C" such that for each £ G 3D we have Lç = L\ e £ç . DEFINITION 7. lei L Ç 3D x Cm+n be a generating CR-bundle over 3D of CR-dimension m whose fibers project isomorphically onto Cm x {0}. We define the partial indices and the total index of the bundle L as the partial indices and the total index of the maximal real bundle C Ç 3D x C". We fix £0 G 3D. Let N(Ç0) be any (m + n) x n matrix whose columns span the real orthogonal complement Lfo ¦ Then the equations of the fibers L^o and Lf are Re(7V*(£0) z w z w 0 = 0 and N*(Q respectively. Here z G Cm and iijgC. Since we are assuming that each fiber of the bundle Lc projects isomorphically onto Cm x {0} Ç Cm+n, the matrix 7V(£0) can be written in the following block form N(to) Go(to) Notto) 80 where N0(Ç0) is an invertible nxn complex matrix. The definition of the bundle C immediately implies that C^o is given by the equations Re(7Vo*(£» = 0. Therefore the columns of the matrix iV0(£0) span the real orthogonal space c± ce. Ço — Let k\, &2, • • •, kn be the partial indices of the bundle C and let A(£) be its characteristic matrix. Then there exists an nxn invertible holomorphic matrix function 0O on D such that the columns of the matrix function Ao(o := e0(e)A0(o (e g öd) span the fibers of the maximal real bundle C. Here A0 denotes the square root of the characteristic matrix A. Once A0 is fixed, there is naturally given basis of the bundle L-1, namely, there is an (m + n) x n matrix function iV(£), £ G 3D, whose rows are from the space £a, whose columns for each £ G 3D span Lj-, and such that K = lA-1 . Let F : D —> <ßm+n ^e an analytic disc with boundary in the generating CR-bundle L. With respect to the splitting of the space Cm+n the mapping F is written as (f,g), where / and g are holomorphic maps into Cm and Cn, respectively. Since for each £ G 3D the matrix function A0(£) is invertible, the 81 mapping g can be written in a unique way in the form g = A0(u + i(v + iTav)) (u,ve£a). Since F has boundary in the bundle L, the discs / and g satisfy the equation MG*o(0f(0 + n*0(09(0) = o (e e 3D) . Hence, since N*A0 equals to iln, Re(G:(0/(0 + *(«(£) - (?>)(£)) - «(0) = 0 (Ç G 9D) and the mapping v is given by the equation (20) v = Re(G*J) . If the partial indices of the bundle C are all greater or equal to —1, then the product extends holomorphically to D and so, given a holomorphic disc / in Cm, the equation (20) is also a sufficient condition for the existence of a holomorphic disc g in Cn such that the disc F = (/, g) has boundary in the bundle L. Even more, in this case one can find an explicit parametrization of all holomorphic discs attached to the bundle L with the parameter space R"+fc x (A°'a(dD))m, where k is the total index of the bundle C Namely, for each holomorphic vector / G (A°'a(dD))m and for each real vector function u G £a such that 82 A0u extends holomorphically to D, there exists exactly one holomorphic disc F = (/, g) attached to L. Before we consider the nonlinear case let us make a few remarks. 1. Since our choice of a change of coordinates in Cm+n involves quite a lot of freedom, it looks like that one could get, using a different change of coordinates, also a different set of attached discs to the bundle L. It is quite easy to construct an example, e.g., the real normal bundle LL C dD x C2 is given by the matrix iV*(£) := [£, £ ], where different linear changes of coordinates result in different sets of partial indices of the associated bundle C. But, as already the above argument shows, as soon as all partial indices of the bundle C are greater or equal to —1, we know how to parametrize all holomorphic discs in Cm+n with boundaries in L. Also, the following simple lemma is true. LEMMA 11. The Banach spaces X = (A°>a(dD))m and R2 x X are naturally isomorphic. Proof. We define l:R2xI —> X as *M,/)(0:=(s + tf)+£/(0 (^dD). It is easy to verify that ^ is one to one and onto bounded linear map. H 83 Since the parameter space of holomorphic discs attached to L is Wl+k x (A°'a(dD))m, we get that as soon as the CR-dimension of the bundle L is at least 1, the set of parameters is isomorphic either to (A°'a(dD))m or to M x (A0,a(dD))m, depending on the codimension n and the orientability of the bundle L. Observe that L is orientable if and only if the bundle C is orientable and thus L is orientable if and only if the total index k is an even integer. Moreover, in the example of the CR-bundle L C dD x C2, where its real normal bundle L1- is given by the matrix iV*(£) = [£ ,£2], one can see that it can also happen that a certain linear change of coordinates can produce only positive partial indices but some other only negative partial indices. Therefore, to work on general CR-manifolds, we will have to assume that there exists a linear change of coordinates in Cm+n such that in the new coordinates each fiber of the maximal complex tangent bundle along a certain curve projects isomorphically onto Cm x {0} and the corresponding partial indices are all greater or equal to —1. Observe that in the case of positive CR-dimension the condition that all partial indices are negative does not necessary imply, as in the case of maximal real bundles, that there is no nearby analytic discs attached to L. See the next remark. 2. The set of discs attached to a generating CR-bundle of a positive CR-dimension is always parametrized by an infinite dimensional Banach space. Even in the case where all partial indices of the associated bundle C are negative we 84 will find a subspace of finite codimension in (A0,a(dD))m which is in one to one correspondence with the analytic discs with boundaries in L. As it was already seen above, a necessary condition for a disc (/, g) in Cm+" to be attached to the bundle L is v = Re(G*J) . To get all holomorphic discs attached to L the function v should be such that there exists a mapping u G £a such that A0(u + i(v + iTav)) is the boundary value of a holomorphic disc in C". For each partial index k j > —1 the dimension of the corresponding set of parameters is k j + 1. But if k j < —1, then the function u j has to be chosen to be identically 0 and the sufficient condition on Vj to generate a holomorphic disc is that the Fourier coefficients t>}(0), Vj(l),..., «}([ 2 ] — 1) are all equal to 0. Here [x], x G M, stands for the greatest integer less or equal to x. This condition is equivalent to the condition j-2-K Fjs(f)= / eÏS0Re(ujJ(9)-f(6))d6 = O, Jo for s = 0,1,..., \\kj\ — 1 in the case k j is an even integer and for s-1-3- h-i 2'2''"'2J in the case kj is an odd integer. Here ujj stands for the j-th row of the matrix G*0. Since the linear functionals FjS are continuos on the space (A0,a(dD))m in 85 the case kj is an even integer and on the space (^4°>a)m in the case kj is an odd integer, the claim is proved. 3. The following example shows that in the nonlinear case some assumptions on the partial indices are really needed. We already know that this can happen in the maximal real case, but when the CR-dimension is positive the difference can be even more striking. Namely, although in the linear model the set of solutions is always parametrized by an infinite dimensional vector space, it can happen that the set of local nearby solutions on a CR-manifold is only finite dimensional. Example. Let Mç := {(z,w) G C2;Im(Çw) = \z\2} (£ G d D) . Then the disc e ^(0,0) (ÇedD) is the only analytic disc with boundary in the fibration {M^}^edD. Proof. Let (f,g) be an analytic disc with boundary in the fibration {M^}^egD. Then M&(0) = l/(0I2 (£eöD). But /*27T /*27T o=/ im(tg(t))de= \f(0\2de Jo Jo 86 and so / = 0. But then Im(£0(O) = 0 on dD and so also g = 0. H Observe that in the above example the matrix A(Ç) equals to (0, i£) and thus the only partial index is —2. One can also define a 4-dimensional submanifold of C3 of CR-dimension 1 with a similar property. Let M := [J {£} x Mç . iedD Then any holomorphic disc with boundary in M and close to the disc e^(e,o,o) (tcdD) is of the form e^(a(0,0,0) (ÇGdD), where a is an automorphism of the unit disc close to the identity. Thus the family of such discs is 3-dimensional. H We consider now the nonlinear case. Let {M(Ç)}çedD be a family of generating CR-submanifolds of CR-dimension m in Cm+" and let p : 3D —> Cm+n be a map of class C0,a such that Pit) e M(0 (£ e dD) . 87 We say that the family {M(Ç)}çegD is a C0,a generating CR-fibration over the unit circle 3D with C2 fibers if for each £0 G 3D there are a neighbourhood Uç„ Ç 3D of £0, an open ball B^o C Cm+n centered at the origin and maps Pi ,..., pf° from the space C0,a(Uço, C2(B^o)) such that for every £ G Uç0 1. the CR-submanifold M(£) n (p(£) + 5ç0) equals to {(z,w) G p(£) + Bio; pf (£, (z, w) - P(0) = 0, j = 1,..., n} , 2. pf(£,0,0) = 0, j = l,...,n,and 3. d^p?" (£, 2, u>) A • • • A 3z,wpÌ° (Ç, z,w)^0 on %. THEOREM 3. Let M(£) Ç Cm+n, Ç G 9D; 6e a C°'a generating CR-fibration over the unit circle 3D with C2 fibers and CR-dimension m. Let p:3D —> Cm+n be a C0,a closed path in Cm+n such that Pit) e M(0 (£ e 3D) . Assume that there exists a linear change of coordinates in Cm+n such that in the new coordinate system all maximal complex subspaces of the generating CR- bundle L := (J {C} x TmM(0 iedD project isomorphically onto the subspace Cm x {0}. Assume also that all partial indices of the corresponding maximal real bundle C Ç 3D x Cn are greater or equal to —1 and that the total index is k. Then there are an open neighbourhood 88 U of 0 E Wn+k, an open neighbourhood V of the function 0 in (A°'a(3D))m, an open neighbourhood W of p in (C0,a(3D))m+n and a map tf :U xV —> W of class C1 such that 1) #(0,0) =p, 2) for each (t, f) E U x V the map p := #(t, /) — p extends holomorphically to D and is such that p(Ç) E M(£) for each Ç G 3D, 3) #(ti, /) ^ #(^2; /) /or ti ^ ^2 /rom £/ie neighbourhood U and any f E V, 4) if p E W satisfies the condition p(£) G M(£), £ G 9-D, and is suc/i £/ia£ p —p extends holomorphically to D, then there are t E U and f EV such that ^(t,f) =p. Proof. Since we are assuming that all maximal complex subspaces project isomorphically onto the subspace Cm x {0}, one can, using the same construction as in Lemma 6, find a set p(£,z,w) = (pi(£,*,u>),...,pn(£,z,u;)) of "global" defining functions for the fibration {M(Ç)}çe9o, i-e., there exist an r0 > 0 and functions ^eC£a(9AC2(Bro)) (!<3a(3D))m x£ax8a^£a by *(/,«,t;)(0:=p(e,/(0>^(« + ^ + ^i;))(0) (ee9D) . Then for every v G £a and £ G 3D we have (A,tf(0,0,0)i;)(O = 2Re(dwp(Çi0,0)Ao(t)i(v(Ç) + i(T<7v)(0)) = -2v(Ç) and thus the partial derivative of the mapping \t with respect to variable v is an invertible linear map from the space £a into itself. By the implicit mapping 90 theorem one can find a neighbourhood V of the zero function in (A0,a(dD))m, neighbourhoods W and U of the zero function in £a and a unique mapping iß:ÜxV—>W such that a triple (f, u, v) E V x U x W solves the equation (21) *(f,u,v) = 0 if and only if t> = tp(u, f). Finally one would like to select from the above family of all possible C0,a closed curves in the CR-fibration {M(Ç)}çedD near p those which bound a sum p + analytic disc . The rest of the argument is the same as in the proof of Theorem 1. At this point one should assume that all partial indices of the maximal real bundle C are greater or equal to — 1. In this case the vector function A0(v + iTav) extends holomorphically to D. This follows from the fact that for any odd partial index kj the function Vj + iTvj is of the form r(Ç)g°(Ç) for some function g° G A°'a(dD). We recall that r(£) represents the principal branch of the square root v^- So the condition on the vector function C .—? A0(u + i(v + #»)(£) (£ E 3D) 91 to extend holomorphically into D is in the case where k j > —1, j = 1,..., n, equivalent to the condition that the vector function e ^ A0(t)u(o (e g 3D) extends holomorphically to D. To find all such functions u G £a one has to find all vector functions a G (A0,a(dD))n such that on d D Aä = a , i.e., for all j = 1,..., n ^^(0 = ^(0 (ÇedD). For any partial index k j = — 1 the only solution of the above equation is a j = 0 and for kj > 0 one has a kj + 1 dimensional parameter family of solutions. Hence, altogether one gets a k + n parameter family of solutions. H The rest of this section was inspired by the work [Bao-Rot-Tre] by Bao-uendi, Rothschild and Trepreau. See also the paper [Turn] by Tumanov for some related results and definitions. We recall the definition of the conormal bundle of a CR-submanifold M C C^ as given in [Bao-Rot-Tre]. We identify the complex bundle A1'0^ of (1, 0) forms on CN with the real cotangent bundle T*CN as follows. To a real 1-form r = ^2 Cjdzj +cjd~žj on C^ we associate the complex (1, 0) form 7 = 2i ^2 cjdzj 92 so that the pairings between the vectors and covectors are related by the identity (r,X)=Im(7)X) for all X G TZCN. Under this identification, the fiber of the conormal bundle E(M) on a CR-submanifold M at the point p E M is given by £p(M) = {7 G A1'°CJV;Im(7,X) = 0,X G TpM} . If the manifold M is generating, then the conormal bundle can be naturally identified with the characteristic bundle (TCM)-L of the CR-structure on M. If locally, near some point p G M, the submanifold M is generating and is given by the set of equations p = (pi,..., pn) = 0, then the fiber of the conormal bundle over the point p is given by SP(M) = {is^pip) = % J2 Sj^ip^Sj G R, 1 < j < n} . j From now on let M(£) Ç Cm+n, £ G ÖD, be a generating CR-fibration over the unit circle dD of class C0,a with C2 fibers and with CR-dimension m. Let p : 3D —> Cm+n be a C0,a curve such that P(0 e M(0 (£ e ÖD) . 93 Let Vp be the set of all holomorphic discs c(£) = (ci(£),..., cTO+n(£)) of class C0,a such that for each £ G 9D the (1,0) form m+n (22) Eci(0^i i=i belongs to the space EP(£)M(£). For each £ G <9D we denote by V^(£) Ç Ep(£)M(£) the subset consisting of all such forms (22), Vp(0 = {1e E,(0M(0;7 = J>(^,c G Fp} . Clearly VJ,(£) is a real linear subspace of Ep(£)M(£). Henceforth we will assume that the coordinates in Cm+n can be chosen so that each maximal complex subspace of the tangent space Tp^M(£), £ G <9D, projects isomorphically onto Cm x {0}. We recall that this is always possible in the case the fibration M(£) Ç Cm+", £ G 3D, is of at least class C^i.e., the defining functions of the fibration belong to the space C1(9D, C2(_Bro)) for some r0 > 0, and the closed path p is of class C1. We also recall that for each £ G 9-D the columns of the matrix function N(0 No(0 span the fiber of the normal bundle of the submanifold M(£) at the point p(Ç). The following characterization of elements of VP(Ç), £ G 3D, is immediate, see also [Bao-Rot-Tre], Proposition 3.6. 94 PROPOSITION 7. A covector 7 = Y,cidzi e Tp(nM^o) belongs to ih e subspace VP(Ç0) if and only if there is a real function s = (si,..., sn) G £a such that 1. ê = [ci,..., cm+n] = îs*[G*, N*0](Q and 2. ^e covector function sl[G*01 N^\ extends holomorphically to D. Remark. For any real vector function s G £a for which property (2) holds we will say that it generates an element from Vp. COROLLARY 9. If all partial indices of the associated maximal real fibration C are greater or equal to 1, then Vp = {0} and so each of the subspaces VP(Ç), £ G 3D, is trivial. Proof.(Corollary) Let A0 denote the matrix function whose columns for every £ G 3D span the fibers of C. Then A0 = OA0, where O is an invertible holomorphic matrix on D and A0 is the square root of the characteristic matrix A of the maximal real vector bundle C Then N*0 = K®-1 and a necessary condition to get an element from Vp is that there exists a real function s G £a such that slT0 extends holomorphically to D. But since all partial indices of A are greater or equal to 1, one concludes that s has to be 0. H 95 Since our method gives all nearby analytic discs of class C0,a attached to the CR-fibration M(£) Ç Cm+", £ G dD, only in the case when all partial indices of the associated maximal real bundle are greater or equal to —1, this will be the case we will consider from now on. In this case we have already proved, Theorem 3, that the family of all nearby holomorphic discs, i.e., all holomorphic discs F G (A0,a(dD))m+n with the property that the disc p + F is attached to the fibration M(£), £ G dD, forms a Banach submanifold A of the Banach space (A0,a(dD))m+n. In the case where the CR-dimension of the fibration is 0, this submanifold is of finite real dimension n + k, where k is the total index of the fibration, but in the case of positive CR-dimension we get an infinite dimensional submanifold. Also, differentiation of the equation (21) with respect to u and / at the point (0, 0) yields {DuiP)v = 0 and (Dfiß)f = Re(G*J) . We recall that v = ip(u, f) is the solution function of the equation (21) which we got using the implicit mapping theorem in an appropriate Banach space. Thus all vectors of the tangent space T0A to the submanifold A at the point 0 are of the form (f,A0(u + i(v + iTav))) , where / G (A0,a(dD))m, and the functions u, v G £a are such that v = Re(G*/) and A0u extends holomorphically to D. 96 Remark. In the case considered by Baouendi, Rothschild and Trepreau in [Bao-Rot-Tre] one works only in a neighbourhood of a point on a given CR-submanifold and so all partial indices of any nearby holomorphic disc attached to the manifold are 0. It is easy to see that all partial indices of a constant map are 0. On the other hand, this condition is stable under small perturbations of the disc, see [Vek2]. PROPOSITION 8. The dimension of the subspace VP(Ç) Ç Ep(f)(M(£)) does not depend on £ G 3D, i.e., it is the same for every £ G 3D. Remark. This proposition extends the Proposition 3.6 from [Bao-Rot-Tre]. Proof. We split the space Rn into three subspaces R" = R"1 © Rn° © R"-1 , where ri\ is the number of positive partial indices, n0 is the number of partial indices which equal to 0, and n_i is the number of partial indices which equal to — 1. With respect to this splitting we denote the coordinates on Rn by (q,y,t). We recall that every element of the space Vp is of the form ìs*[g*0,n:] for some real function s G £a. We also recall that TV* = %A~X. Since the first n\ partial indices are positive, any real vector function s from the space £a which generates an element in Vp, must have, by the same argument as in the proof 97 of Corollary 9, the first ri\ coordinate functions identically equal to 0. Because of this reason, and to simplify the notation, we will assume, and we can do so without loss of generality, that ri\ = 0. Each element of Vp is now generated by a real function of the form where y G W1", u G Cn_1 and r(£) is the principal branch of the square root. Let k0 be the dimension of the space Vp(l). We will prove that for each £ G 3D the dimension of the space Vp(£) is also k0. Since for each £0 G 3D there exists an automorphism of the unit disc D which takes 1 to 1 and £0 to — 1, it is enough to prove the above claim for £0 = — 1. Let (yj, Re(cjjr(£))), j = 1,..., k0, be a set of real functions on 3D which for each j generate an element of Vp, and such that the real vectors (î/j,Re(o>j)) (j = l,...,k0) are linearly independent. If also the set of vectors (î/j,Re(io>j)) (j = l,...,k0) is linearly independent, the claim is already proved and we are done. Let us assume now that this is not the case and that these vectors are not linearly independent. Then there are real numbers Ai,..., Xk0, not all equal to 0, such 98 that Ko J2 XM = ° i=i and Ko ^AiRe(^) = 0 . i=i The second equation is equivalent to Ko for some real vector t from Rn_1. The way how t is defined immediately implies that t ^ 0 and that the real vector function (0,Re(*r(£))) generates an element from Vp. Since t is a real vector, both functions (0,Re(tr(£))) and (0, Re(iir(£))) generate an element from the space Vp. This follows from the following claim. Claim. Let f = u + iv, u,v E (£ffi'a)", be a vector function such that the function £ ^ Re(r(0)/(0 (ÉG0D) extends holomorphically into D. Then / G (*4°'a)n. In particular, also the function £ ^ Re(ir(0)/(0 (£ed£>) extends holomorphically into D. 99 Proof.(Claim) Since the function £ ^ Re(r(0)/(0 (tCdD) extends holomorphically into D, all its negative Fourier coefficients have to vanish. This implies that for every j G N we have fH) + fH - 1) = 0 • Since we also have lim f{-j) = 0 , j->oo we conclude that all negative Fourier coefficients of the function / are 0 and the claim is proved. H Also, since not all real numbers Xj, j = 1,..., k0, are 0, we may assume, without loss of generality, that Ai ^ 0. We repeat the above argument on the set of real functions (?/j, Refu^r(£))), j = 2,...,k0, and the vector function (0, Re (iir (£))). If at £ = — 1 these vectors are still linearly dependent, one can find real numbers Ai,..., A^o, not all equal to 0, such that Ko X\it + 2_. ^j^j = ^i for some nonzero real vector t\ G Rn_1. We observe that it can not happen that A2 = • • • = Xh0 = 0 since the vectors t and t\ are real. We also observe that t and t\ are linearly independent vectors. Repeating the above argument 100 we either stop at the jth step, j < k0, or we produce k0 linearly independent vectors which span Vp(—1). M So we can define the defect of the closed curve p in a generating CR-fibration over dD with partial indices greater or equal to —1 in the same way as Baouendi, Rothschild and Trepreau do in [Bao-Rot-Tre]. See Definition 3.5 and Proposition 3.6 in [Bao-Rot-Tre]. See also [Turn]. DEFINITION 8. The defect del (p) of the curve p is defined as the dimension of the real vector spaces Vp(£), £ G dD. From now on we will restrict our discussion to the set A* of holomorphic perturbations of p which leave one of the points, say p(l), on the curve p fixed. But to prove that the set A* is in fact a manifold, we have to assume that all partial indices of the path p are nonnegative. See the examples at the end of this section. Let £„,* Ç £a and {Al'a(dD))m Ç (A°>a(dD))m be the subspaces of the functions which are 0 at Ç = 1. Let T, : C^idD) -+ (%a(dD) be the Hilbert transform which assigns to a function v E C¥^a(dD) the harmonic conjugate function v for which v(l) = 0. Since T* does not preserve the subspace of odd functions in CR'a(<9D), there is no natural way of defining an appropriate Hilbert transform on £a^. 101 Let k be the total index of the associated maximal real bundle C Ç dD x Cn. Then the following lemma holds. LEMMA 12. A* is a Banach submanifold of the manifold A of the infinite dimension in the case the CR-dimension of the fibration {M(Ç)}çegD is positive, and of real dimension k in the case of maximal real fibration over dD. Proof. We define the map F : (A°^(dD))m x £^ x £^ —? £^ by F{f,uiV){i) := p&fi&Aodü + Trv) + i(v + iTav))(Ç)) (^ e dD) . Here p = (pi,..., pn) is the set of defining functions of the fibration {M(Ç)}çegD along the path p. Using the implicit mapping theorem as in the proof of Theorem 3 one gets a neighbourhood M of the zero function in (A®,a(dD))m, neighbourhoods U and V of 0 in £a^, and a unique mapping -0 : Af x U —> V such that the triple (f,u,v) G M x U x V solves the equation F(f, u, v) = 0 if and only if U = 1p(f,Ü). As we already know a necessary and sufficient condition for any disc from the above family to be the boundary value of a holomorphic disc is that the mapping A0(ù + Tav) 102 extends holomorphically to D. Let \t(£), t G R"+fc, denote the linear parametri-zation (7), (8) of all real functions u G £a such that A0u extends holomorphically to D. Thus to extract from the above family of discs A0(u + iip(f,u)) all holomorphic discs which are 0 at £ = 1, we have to find all functions ù G £CT;* and values t G Wn+n which solve the equations u + Ttri>(f,v) = *(t) and Ttri>(f,v){l) = *(t){l). Since all partial indices are nonnegative, the nx (n + k) matrix Dt(^(t)(l))\t=0 has the maximal rank. Since we also have Dy{u + Tff^(/, u)) |/=o,ü=o = /rf > one gets, using the implicit mapping theorem again, a unique mapping /j, from a neighbourhood of the point (0,0) G (A^a(V))m x Rfe into £CTj* such that all small holomorphic disc (/, g) from (A®,a(dD))m+n which solve the equation p(e,/(O>0(O) = o (^9D) are of the form (/,A0(/i(/,s) + #(/,/i(/,s)))) for a unique pair (/, s) from a neighbourhood of the point (0, 0) in (^4°'a(D))m x Rk. m Note that any element of the tangent space T0A* is of the form (f,A0(u + i(v + iTav))) 103 for some / G (A®>a(dD))m, v G £CTj* such that v = Re(G*0f), and u G £a such that A0u extends holomorphically to D and for which one also has u—Tav G £a^. Henceforth our goal will be to reprove and to generalize Theorem 1 from [Bao-Rot-Tre]. In fact we can prove the same statement for an arbitrary closed path p in a generating CR-fibration M(£) C Cm+", £ G 3D, for which there exists a linear change of coordinates in Cm+" such that the partial indices of the corresponding maximal real bundle are all greater or equal to 0. We recall the definition of the evaluation maps JFç defined on the manifold A*, see [Bao-Rot-Tre] for more details. See also [Turn]. For every £ G 3D and F G A* we define ^(F):=(p + F)(0. Then for every £ G 3D the derivative ^(0) maps the tangent space TÜA* into TP(t)M(Q. THEOREM 4. Let p and M(£) Ç Cm+n, Ç G 3D, be as above. Then for each £ G 9-D, £ ^ 1, one has (23) ^(0)(T0A) = Vp(O-L. Proof. We first prove the following partial statement, namely, 104 for each £ G 3D. To prove this claim let (24) (f,A0(u + x(v + xTav))) be an arbitrary element of T0A*. We recall that / G (A®'a(dD))m, that u G £CT, that v,u — Tav G £CT;*, and that also v = Re(G*J) . On the other hand let (25) <[Go,K] be an element of Vp. Here u0 G £a. Since both vector functions (24) and (25) extend holomorphically to D, their product also has to extend holomorphically to D. But on the other hand the multiplication of (24) and (25) yields a purely imaginary vector function m*(w - Tav + lm(G*J)) . Since the vector function (24) is 0 at £ = 1, the claim is proved. To prove that in the above inclusion in fact the equality holds it is enough to prove that the dimension of the space JFç(0)(T0^4i(i) is 2m + n — def(p). Since for each £ G 3D, £ ^ 1, one can find an automorphism of the unit disc D which takes 1 to 1 and £ to —1, it is enough to prove the claim for £ = — 1. For this it is enough, since the set of function values {/(—1); / G (A®'a(3D))m} already 105 spans a 2m-dimensional subspace, to prove that the subspace {(u - 7»(-l); v = Re(G:/), /(-1) = 0, A0u G {A^a{dD))n, u - Tav G £a,,} has dimension n — def(p). Denote by n\ the number of positive indices and by n0 the number of indices which are equal to 0, and split the space Rn correspondingly. For each positive partial index kj the set of real functions Uj such that the function £ i-» £fci/'2%(£) extends holomorphically to D and (uj — Tvj)(1) = 0 is at least 1-dimensional. Thus the proof of the claim will be finished once we prove that the following subspace of Rn° {((U-Tcrv)(-iy,v = Re(G:f),f(-l) = 0,u-TcrveSa,*}f)Rn° has dimension n — ri\ — def(p). To prove the last claim we will show that for a vector u0 G M"° the condition uto{u-Tav)(-l) = 0 for every v G (Cffi'a((9D))no such that v is given as the last n0 component functions of Re(cr*/), /(—1) = 0, and every constant vector u G IR™0 such that (u — TŒv)(l) = 0, implies (oX)[g;,ack-i)gvu-i). This will complete the proof of (23). But since every real vector function ù0 G £a which generates an element from Vp has the first ri\ coordinate functions 106 identically equal to 0, see the proof of Corollary 9, it is enough to prove the statement for the case where all partial indices are 0 and n\ = 0. From here on the argument goes very much the same as the one given by Baouendi, Rothschild and Trepreau in [Bao-Rot-Tre]. We recall that T* denotes the Hilbert transform on (A^a(dD))n such that for every v G (A^a(dD))n we have (7»(1) = 0. Also, since all partial indices are 0, the vector function u is in fact a constant such that (Tav — u)(l) = 0 and hence Tav — u = T*v . Let u0 G M"° be a vector with the property that <(7>)(-l)=0 for every v G (C$a(dD))n such that v = Re(G*f), /(-1) = 0. We recall that for v G C^a(dD) and £0 G dD one has 2tt i" Jo (£-!)(£-&>) where £ stands for e%e. We denote the vector function ul0G*0 by a*. Then for every nonnegative integer q, every vector z0 G Cm, and a function / of the form /(e) = (e2 -1)^ (eeöD) 107 one has (26) WT.v))(-l) = PV^ [2W 2j0§^T^de (27) = —J [C^afâzo-p+ialfàzolM . By our assumption the integrals (26) and (27) equal 0 for every nonnegative integer q and every vector z0 G Cm. Since one can also take iz0 instead of z0, one gets that 2i 27 for every nonnegative integer q. The above identity can be written in terms of Fourier coefficient as a0{-q-1) = 0 (q = 0,1,2,...) which immediately implies that the real vector ul0 generates an element of Vp. The identity (23) is proved. H For the next theorem we have to assume more regularity on the fibration {M(£)}çeaD and the closed path p. We assume now that we have a C1,a fibration with C3 fibers, i.e., the fibration is given by a set of real functions from the space C1,a(dD, C3(Bro)), and the closed path p shall be of class C1,a. Under this conditions one can repeat the proofs of Theorem 3 and Theorem 4 in the C1,a category. We recall the definition of the mapping Q from [Bao-Rot-Tre]. zyi çq+Lal(Ç)de = 0 108 Let be defined by G : T0 A —> C Ç(F) := j^(e»; im+n e=o THEOREM 5. Let p and M(£) Ç Cm+n; Ç G 3D, be as above. Then Q maps TqA* into Tp(i)M(l) and (28) ö(ToA) = ì^(l)± • Proof. We first observe that for every F G T0A* one has M[G*0, N*0]F) = 0 on 9D. Differentiation with respect to 9 and setting 9 = 0 implies that Q maps T0A, into Tp(i)M(l). The proof of (28) is quite similar to the proof of (23). The inclusion ö(ToA) Ç ì^(l)± follows as above since the product of any two functions G G Vp and F G T0.A* equals to 0, (29) GlF = 0 . Namely, the differentiation of (29) with respect to 9 and setting 9 = 0 yields d G\l)^F{e^ 0 . 0=0 109 To prove the opposite inclusion in (28) we proceed similarly as in the proof of (23) and reduce the problem to to the case where all partial indices of the associated maximal real bundle are 0, and showing the following claim. Claim. The vector space ¦v = Re(G*of),f(Ç) = (É - inqZo,q G NU {0},zo G Cm} {^(7>)(e*0) 0=0 has dimension n — def(p). Proof.(Claim) We are using a similar notation as in the proof of (23). Let u0 G 1" be a real n-vector which annihilates the above vector space. Also, let a* be the vector «*G*. We recall that u°dë{T*v){e )ìe=0 - nj0 (e-if id0 i r2lT _____________ = - / K(mq+1Zo + ai(0¥+1zo]d9. TT JO Replacing z0 by iz0 and adding the identities one gets j-2-K / al(Oe+1d6 = 0 Jo for every nonnegative integer q. In terms of Fourier coefficients we have a0(-q - 1) = 0 for every çGNU {0}. Thus the real vector u0 generates an element from Vp. This finishes the proof of the claim and so also the theorem. H Remarks and examples. In the case when the partial indices of the path p in 110 the generating CR-fibration {M(^)}^egD are not all nonnegative, the conclusions in Theorems 4 and 5 are not true even if we consider only the case where all indices are greater or equal to —1. One problem, of course, occurs if the total index k happens to be negative. Then the number of free parameters is strictly less than the number of additional equations we have to satisfy. Here we give two examples in C2 for which k > 0 but the conclusions of Theorems 4 and 5 still do not hold. Example 1. In this example we find a maximal real fibration in C2 for which the set of attached discs passing through the point (0, 0) does not form a manifold. Let the maximal real fibration {M(^)}^egD be given by the set equations lm(zl) = 0 and Re(wr(0) = Re(r(Ç))Re((zë)2) • It is easy to check that the partial indices of the path p(£) = 0, Ç G 3D, are 2 and —1, and so the total index k equals 1 and the defect of the path p is 1. Hence the dimension of the spaces is also 1. Claim. The family of holomorphic discs with boundaries in the maximal real Ill fibration {M(Ç)}çegD which all pass through the point (0, 0) at £ = 1 is not a manifold. Proof.(Claim) Let (z, w) be a holomorphic disc with boundary in the maximal real fibration {M(Ç)}çedD and such that (z(l),w(l)) = (0,0). Then from the first equation Im(z(£)£) = 0 we get *(£)£ = w£-2Re(u;)+ûJë for some complex number u. The second equation Re(w(0r(0) = Re(r(Ç))Re((z(e)ë)2) (£ E 3D) implies Re(w(e)0 = Re(Ç)Re((z(e2)f )2) (£ G ÖD) . A short calculation shows that the right hand side of the last equation equals to Re(w2Ç5 + (lo2 - 4coRe(co))^ + (2|w|2 + 4(Re(u>))2 - 4o>Re(u;))Ç) . Thus W(Ç) = u)2(l + (u2 - 4u>Re(u>))£ + (2\co\2 + 4(Re(u;))2 - 4a>Re(u;)) and so one must have u2 + (to2 - 4wRe(u;)) + (2\cj\2 + 4(Re(u;))2 - 4u;Re(a>)) = 0 112 or after division by 2 cj2 - 4uRe(cj) + \u\2 + 2(Re(u>))2 = 0 . If we write u = x + iy, then the imaginary part of the last equation yields -2xy = 0 . Thus the constant u has to be either real or purely imaginary. So the set of solutions of the above equations is the union of two intersecting curves in (A0,a(3D))2 and therefore not a manifold. H Example 2. Let the maximal real fibration in C2 be given by ka(zr(gj) = 0 and Im(u>r(£)) = Re((zr(ë))3) . Then the partial indices of the closed path p(£) = 0, £ G 3D, are 1 and —1 and so the total index is 0. Also, the defect def(p) is 1 and thus the dimension of the spaces V^-(^), £ G 3D, is 1. Let (z,w) be a holomorphic disc with boundary in the maximal real fibration {M(Ç)}çegD and such that (z(l),w(l)) = (0,0). 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