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STATIONARY DISCS GLUED TO A LEVI NON-DEGENERATE HYPERSURFACE LEÁBLANC-CENTI 8 Abstract. Weobtainanexplicitparametrizationofstationarydiscsgluedto 0 some Levi non-degenerate hypersurfaces. These discs form a family which is 0 invariantundertheactionofbiholomorphisms. Weusethisparametrizationto 2 constructalocalcircularrepresentationofthesehypersurfaces. Asacorollary, n wegettheuniquenessofbiholomorphismswithgiven1-jetatsomeconvenient a point. J 1 1 A central problem in complex analysis consists in classifying the domains in Cn under the action of biholomorphisms. If n = 1, the Riemann mapping theorem ] V states that every simply connected domain of C, which is not all of C, is biholo- C morphic to the unit disc. We know since the work of H. Poincaré [10] that this . theorem has no immediate generalization in the case n ≥ 2. Indeed, the rigidity h of the holomorphy condition in the multidimensional situation makes biholomor- t a phic equivalence very rare. This leads us to seek biholomorphic invariants of the m boundary of a domain. S.S. Chern and J.K. Moser [3] have associated to any Levi [ non-degeneraterealhypersurfaceasimpleequationandproposedapurelygeometric construction of classifying invariants. Stationary discs are another natural invari- 2 v antofmanifoldswithboundarywithrespecttobiholomorphisms,ormoregenerally 1 with respect to CR maps. 5 WerecallthataholomorphicdiscinCn isaholomorphicmapfromtheunitdisc 1 ∆ in C to M, continuous up to the boundary. L. Lempert [8] showed that some 2 ofthesediscs haveinterestingextremalitypropertieswith respectto the Kobayashi 1 6 metric. Indeed, in a strongly convex domain in Cn, the geodesics for this metric 0 are exactly the stationary discs. Using this particular family of discs, L. Lempert / constructed a homeomorphism between any strongly convex domain and the unit h t ball,whichisananalogueoftheRiemannmap. Moregenerally,theparametrization a of M by the stationary discs yields to a (local) representation of M as a circular m submanifold. Inhighercodimension,A.SukhovetA.Tumanov[11]obtained,using : v the same method, a new invariant of small perturbations of the product of spheres i S3×S3, canonically diffeomorphic to the conormal bundle. The map constructed X commuteswithbiholomorphisms,andisapartialanalogueofthe circularrepresen- r tation given by L. Lempert. a The aim of this paper is to give some explicit parametrization of the stationary discs glued to small deformations of the hyperquadric Q defined by 0=r(z)=ℜz − t′z¯A′z, 0 where A denotes a n-sized non-degenerate hermitian matrix. More precisely: 1 2 LEÁBLANC-CENTI Theorem 1. Pick p = (p ,0,...,0) ∈/ Q and let h0 be a stationary disc glued to 0 Q and centered at p. Then, for any ρ near r with respect to the C3-topology, the set of all non-constant stationary discs glued to the hypersurface {ρ = 0} forms a (4n+3)-parameter family near h0. If we only consider the discs h centered at p, the maps h7→h(1) and h7→h′(0) are local diffeomorphisms on their images. Note that Theorem 1 shows that a hypersurface “near” Q is represented in a neighborhood of p in a circular way by its Kobayashi indicatrix {h′(0)}. The idea of the proof (see [2, 11, 6]) consists in using a criterion of J. Globevnik [7]: given a submanifold E and a disc f glued to E, and under the condition that the partial indices of E along f are non-negative, the holomorphic discs near f glued to a submanifold near E form a κ-parameter family, where κ is the Maslov index. In our case, the main difficulty is to check that partial indices are non-negative, and to compute κ. Using the biholomorphic invariance property of stationary discs, we get as a corollarythat a biholomorphismfixing p in a neighbourhoodof small deformations of Q is (locally) uniquely determined by its differential at p. This paper is organized as follows. We begin with some definitions in the first section. The second section is devoted to the proof of Theorem 1. We then apply this result to the study of biholomorphisms in the third section (Theorem 4). In the Appendices, we prove two quite technical lemmas, used in Section 2. 1. Preliminaries LetM bearealhypersurfaceinCnandρbesomedefiningfunctionofM (thatis, M ={ρ=0}anddρdoesnotvanishonM). Aholomorphic dischisaholomorphic function from the unit disc ∆ in C to Cn, continuous up to the boundary. We say that h is glued to M if h(∂∆)⊂M. We recallthe constructionof the conormalbundle N∗M of M. Its fibre at some pointp∈M isthesetofall(1,0)-formsonTCnwhoserealpartvanishesidentically on the tangent space T M: p N∗M ={φ∈T∗Cn/ ℜφ =0}. p p |TpM Notice that N∗M is a (real) line generated by ∂ ρ. We will need the p p Proposition 1. [12] A real hypersurface M in Cn is Levi non-degenerate if and only if its conormal bundle N∗M is totally real out of the zero section. Definition 1. A holomorphic disc h glued to M is stationary if there exists a meromorphic lift (h,h∗) of h to the cotangent bundle T∗Cn, with at most one pole of order 1 at 0, such that ∀ζ ∈∂∆, h∗(ζ)∈N∗ M \{0}. h(ζ) Remark 1. In coordinates, Definition 1 is equivalent to the existence of some continuous function c:∂∆→R∗ such that h∗(ζ)=c(ζ)∂ρ on ∂∆ and ζh∗ extends h(ζ) holomorphically to ∆. This notion is preserved under the action of biholomorphisms. STATIONARY DISCS GLUED TO A LEVI NON-DEGENERATE HYPERSURFACE 3 Definition1isinfactslightlydifferentfromtheonegivenbyL.Lempert,sincewe assumethattheboundaryofthe liftdoesnotmeetthezerosectionofthe conormal bundle. In[8],evenifthisconditionisnotassumedinthedefinition,itisnecessarily satisfied because of the strong convexity of the domain. Notice that, in light of Proposition 1, the regularity results known for holomorphic discs glued to a totally real submanifold [4] apply to the map (h,h∗). Hence the lift is as regular to the boundary as N∗M. For this reason, following [11], we will call in the sequel (h,h∗) (or h∗, with a minor abuse of notation) a regular lift of h. A very simple computation shows that stationary discs glued to the unit sphere and centered at 0 are exactly the linear discs. This result is also valid for general circular strongly convex domains. Indeed, for circular domains, the linear discs are stationary,andconverselyfor circularstronglyconvexdomains,allstationarydiscs through0arelinearbyLempert’suniquenessresult. Thisprovidestwoparametriza- tions of such discs, via the maps h 7→h(1) and h 7→ h′(0). In the next section, we obtain the same parametrizations only supposing the non-degeneracy of M. 2. Proof of Theorem 1 Consider the hyperquadric Q⊂Cn+1 defined on an open set Ω∋0 by n 0=r(z)=Rez − a z¯z , 0 i,j i j i,j=1 X where the hermitian matrix A=(a ) is non-degenerate. i,j i,j 2.1. Stationary discs glued to Q. Wefirstdetermineexplicitelystationarydiscs glued to the hyperquadric. Proposition 2. Stationary discs glued to Q are exactly under the form ζ tw¯Aw 1+aζ ζ h(ζ)= tv¯Av+2tv¯Aw + +iy ,v+w 1−aζ 1−|a|2 1−aζ 0 1−aζ (cid:18) (cid:19) where v,w ∈Cn, y ∈R, a∈∆. 0 Moreover h∗ is a regular lift of h if and only if there exists b∈R∗ such that b −a¯ a ∀ζ ∈∆\{0}, h∗(ζ)= +ζ− ζ2 ×(1/2,−th (ζ)A). ζ 1+|a|2 1+|a|2 α (cid:18) (cid:19) Remark 2. If the disc h is centered at some point p=(p ,0,...,0), we obtain 0 tw¯Aw 1+aζ ζ h(ζ)= +iy ,w . 1−|a|2 1−aζ 0 1−aζ (cid:18) (cid:19) Proof. Necessary condition Suppose that h has some regular lift h∗, and pick c as in Remark 1: ∂r (1) ∀ζ ∈∂∆, h∗(ζ)=c(ζ)× ◦h(ζ)=c(ζ)×(1/2,−th (ζ)·A). α ∂z Thus, there exists some holomorphic function ϕ on ∆, continuous up to ∂∆ and such that for all ζ ∈∂∆, c(ζ) =ϕ(ζ)/ζ ∈R∗. Expanding ϕ into Fourier series, |∂∆ we obtain that ϕ(ζ)=a+bζ+a¯ζ2 for some a,b∈C, and (2) ∀ζ ∈∂∆, c(ζ)=aζ¯+b+a¯ζ. 4 LEÁBLANC-CENTI • First case: a=0 (and hence b6=0). That is, for all ζ ∈∂∆, h∗(ζ)=b(1/2,−th (ζ)·A). By hypothesis, h∗ has at most α onepoleoforder1at0,henceζh¯ isholomorphicin∆. Thenthereexistv,w∈Cn α such that h (ζ)=v+ζw. Since h is glued to Q, we get by means of an expansion α intoFourierseriesthath isaffine,anduniquelydefineduptotheadditionofsome 0 purely imaginary constant. • Second case: a6=0. Let us denote by a and a the zeroes of 0 = a +bζ +a¯ζ2. Since h∗ does not 1 2 vanish on∂∆, the moduli ofa anda are different from1, and |a a |=|a/a¯|=1: 1 2 1 2 assume for example that 0 < |a | < 1 < |a |. Writing h (ζ) = +∞H ζk in 1 2 α k=0 k L2(∂∆), we get that the map ζh∗ extends holomorphically in ∆ if and only if P ζ 7→(a+bζ+a¯ζ2)h (ζ) extends holomorphically in ∆, that is, α (3) ∀k ≥1, aH¯ +bH¯ +a¯H¯ =0. k k+1 k+2 HencethereexistV,W ∈Cn,dependingonlyonH andH ,suchthatforallk ≥1, 1 2 H = a¯k−1V +a¯k−1W. In fact, it is easy to prove that W = 0, by using the fact k 1 2 that the radius of convergence of each series (v a¯k−1+w a¯k−1)ζk is larger than j 1 j 2 1. Hence H =h (0), H =h′ (0) and 0 α 1 α P +∞ (4) h (ζ)=H +ζH (a¯ ζ)k. α 0 1 1 k=0 X Since a and a are not of modulus 1, we get b 6= 0. By multiplying h∗ by 1 2 1/b, we can assume b = 1. The other regular lifts will be obtained by multiplication by some non-zero constant. We search to express a in terms of a . If 1 1−4|a|2 ≤0, one gets |a |=|a |, which is impossible. Consequently, 1−4|a|2 >0 1 2 −1+ 1−4|a|2 and a = . Setting a = |a|eiθ, we obtain Arg(a) = Arg(a )−π 1 1 2a¯ p −a and |a|= |a1| . Finally, a= 1 . 1+|a1|2 1+|a1|2 Sufficient condition Conversely, suppose that h is given as in (4) (with |a | < 1) and h is uniquely 1 0 determined (up to the addition of some imaginary constant) in order to glue h to Q. If a = 0, h is affine, and the expressions (1) and (2) show that h is 1 α stationary and give its regular lifts. So let us assume that 0<|a |<1, and define 1 c(ζ) = − a1 ζ¯+1− a¯1 ζ. Then h∗ given by (1) is the only regular lift of h 1+|a1|2 1+|a1|2 up to multiplicative non-zero constant. (cid:3) Proposition 3. The map Φ : (y ,v,w,a) 7→ h is a smooth diffeomorphism from 0 R×Cn×(Cn\{0})×∆ to the set of all non-constant stationary discs glued to Q. Its inverse is given by (5) Φ−1 :h7→(ℑh (0),h (0),h′ (0),[θ (1)−iθ (i)]) 0 α α h h ||h′ (0)||2 1 1 where θ (ζ)= α − . h 4 ||h (−ζ)−h (0)||2 ||h (ζ)−h (0)||2 (cid:18) α α α α (cid:19) Proof. Byconstruction,themapΦisontoandsmoothwithrespecttothetopology induced by Cα(∂∆)n+1. Moreover, if h = Φ(y ,v,w,a), then y = ℑh (0) and for 0 0 0 STATIONARY DISCS GLUED TO A LEVI NON-DEGENERATE HYPERSURFACE 5 all ζ ∈ ∆, h (ζ) = v +w ζ +∞(aζ)k . The uniqueness of the development in α k=0 power series shows that Φ ihs injective. i P Let us notice that if h=Φ(y ,v,w,a), then for all ζ ∈∂∆, 0 ||h′ (0)||2 1−aζ 2 α = =1−2ℜ(ζa)+|a|2, ||h (ζ)−h (0)||2 ζ α α (cid:12) (cid:12) which gives Φ−1. In order to get the(cid:12)(cid:12) smoot(cid:12)(cid:12)hness of Φ−1, it suffices to verify that the linear maps h 7→ h (0) and h 7→(cid:12)h′ (0) a(cid:12)re continuous on Cα(∆¯)∩H(∆). But α α this comes from Cauchy inequalities. It remains to prove that Φ is a submersion. As the image of dΦ is of finite dimension at any point, it suffices to prove that dΦ is injective for all (y0,v,w,a) (y ,v,w,a) ∈ R×Cn×(Cn\{0})×∆. Let (y′,v′,w′,a′) ∈ R×Cn×Cn×C be 0 0 such that dΦ (y′,v′,w′,a′)=0. Writing the partial differentials, we get: (y0,v,w,a) 0 0=d Φ y′ +d Φ v′+d Φ w′+d Φ a′. y0 (y0,v,w,a) 0 v (y0,v,w,a) w (y0,v,w,a) a (y0,v,w,a) The n last components of this equality give ζ ζ2 ∀ζ ∈∆¯, v′+w′ +w a′ =0. 1−aζ (1−aζ)2 Hence v′ = w′ = 0 and wa′ = 0, which implies a′ = 0. If we replace in the first component, we obtain y′ =0, which concludes the proof. (cid:3) 0 Remark 3. The set of all non-constant stationary discs glued to Q and centered at p is either empty, or a submanifold of (real) dimension 2n+1. Corollary 1. Pick p ∈ Cn+1. The set M of all non-constant stationary discs p glued to the hyperquadric Q = {r = 0} and centered at p is non-empty if and only if one of the following conditions holds: * A is positive definite and r(p)<0; * A is negative definite and r(p)>0; * A has two eigenvalues of different signs. If so, the map M ∋h7→h(1)∈Q is a local diffeomorphism if and only if p∈/ Q. p Proof. Let h be a stationary disc glued to Q: ζ tw¯Aw 1+aζ ζ h(ζ)= tv¯Av+2tv¯Aw + +iy ,v+w 1−aζ 1−|a|2 1−aζ 0 1−aζ (cid:18) (cid:19) where y ∈R, v,w ∈Cn, a∈∆. In particular, 0 tw¯Aw (6) h(0)= tv¯Av+ +iy ,v ∈Q⇔ tw¯Aw =0. 1−|a|2 0 (cid:18) (cid:19) Consequently,thesetM isnon-emptyifandonlyifthereexistsw ∈Cn\{0}such p that tw¯Aw =ℜp − tp¯ Ap . This gives the three cases. 0 α α Set x = ℜp − tp¯ Ap . According to the previous proposition, it suffices to 0 0 α α consider the map (y ,v,w,a)∈R×Cn×(Cn\{0})×∆/v =p ,y =ℑp , tw¯Aw =x →ψ R×Cn 0 α 0 0 1−|a|2 0 (cid:26) (cid:27) defined by tp¯ Aw tw¯Aw w α ψ(v,w,a,y )=(ℑh (1),h (1))= 2ℑ +2x +ℑp ,p + . 0 0 α 1−a 01−|a|2 0 α 1−a (cid:18) (cid:20) (cid:21) (cid:19) 6 LEÁBLANC-CENTI Its differential at some point (y ,v,w,a) is defined on the tangent space 0 tw¯′Aw+ tw¯Aw′ a¯a′+aa¯′ (0,0,w′,a′)∈R×Cn×Cn×C×/ +x =0 1−|a|2 0 1−|a|2 (cid:26) (cid:27) by dψ (0,0,w′,a′) (y0,v,w,a) = 2ℑ tp¯αAw′ + tp¯αAwa′+ x0 a′ , w′ + wa′ . 1−a (1−a)2 (1−a)2 1−a (1−a)2 (cid:16) h i (cid:17) Since M is non-empty, it is a submanifold of (real) dimension 2n+1, and hence p dψ is an isomorphism on R×Cn if and only if it is injective. Moreover, (y0,v,w,a) tw¯′Aw+tw¯Aw′ +x a¯a′+aa¯′ =0 1−|a|2 0 1−|a|2 dψ (0,0,w′,a′)=0 ⇔  ℑ tp¯αAw′ + tp¯αAw + x0 a′ =0 (y0,v,w,a) 1−a (1−a)2a′ (1−a)2  wh′ + w a′ =0 i 1−a (1−a)2  w′ =−1a−′aw ⇔ (− a¯′ − a′ ) tw¯Aw +x a¯a′+aa¯′ =0  1−a¯ 1−a 1−|a|2 0 1−|a|2  ℑ( x0 a′)=0 (1−a)2  w′ =−1a−′aw ⇔  x0 −1a¯−′a¯ − 1a−′a + a¯1a−′+|aa|a¯2′ =0 .  x ℑh a′ =0 i 0 (1−a)2 In particular,if x0 =0 (that is, if p∈Q), dψ(v,w,a,y0) is not injective. If x0 6=0,we canreplaceinthe third rowa¯′ by (1−a¯)2. Thena′ =0 andw′ =0, whichconcludes (1−a)2 the proof. (cid:3) 2.2. Stationary discs glued to a small perturbation of Q. 2.2.1. Discs glued toasmall perturbation of P(N∗Q). Themethodconsistsinusing a theoremof J. Globevnik [7]: given a submanifold E and a disc f gluedto E, and under the conditions that some integers depending on E and f are non-negative, the holomorphic discs near f glued to a submanifold near E form a κ-parameter family, where κ is the Maslov index of E along f. Since every stationary disc h0 glued to a hypersurface Q has a holomorphic lift ˆh0 glued to the projectivization P(N∗Q) of the conormal bundle of Q, we are going to apply the previous criterion to E =P(N∗Q) and hˆ0. Let us precise Globevnik’s statement. Let f : ∂∆ → CN be the restriction to ∂∆ of a holomorphic disc, and assume that f is of class Cα and B ⊂ R2N is an open ball centered at 0. We suppose that there exist some functions r ,...,r in C2(B) such that dr ∧...∧dr does not 1 N 1 N vanish on B, and M ={ω ∈f(ζ)+B/ ∀1≤j ≤N, r (ω)=0} j verifies f(ζ)∈M for all ζ ∈∂∆. We also assume T(ζ)=T M to be totally real f(ζ) for all ζ ∈∂∆. ∂r i Pick ζ ∈ ∂∆, and denote by G(ζ) the invertible matrix (f(ζ)) . For any ∂z¯ (cid:18) j (cid:19)i,j STATIONARY DISCS GLUED TO A LEVI NON-DEGENERATE HYPERSURFACE 7 matrix A(ζ) whose columns generate T(ζ), each row of G(ζ) is orthogonal to each column of A(ζ): −1 −1 ℜ(G(ζ)A(ζ)=0⇐⇒G(ζ)A(ζ) =−G(ζ)A(ζ)=⇒A(ζ)A(ζ) =−G(ζ) G(ζ). −1 Set B(ζ)=A(ζ)A(ζ) for all ζ ∈∂∆. The matrix B does not depend on A. Moreover,one can find a Birkhoff factorizationof B [1], that is some continuous matrixfunctions B+ :∆¯ →GL (C)andB− :(C∪{∞})\∆→GL (C)suchthat N N ∀ζ ∈∂∆, B(ζ)=B+(ζ)Λ(ζ)B−(ζ) where B+ and B− are holomorphic, and Λ(ζ) = diag(ζκ1,...,ζκN). The integers κ ≥ ... ≥ κ do not depend on this factorization. They are called the partial 1 N indices of B (see [13, 5] for more details). We call the partial indices of M along f theintegersκ ≥...≥κ . Noticethattheyonlydependonthebundle{T(ζ)/ζ ∈ 1 N ∂∆}. The Maslov index, or total index, of M along f is κ= Nκ . 1 j We only recall a slightly simpler version than the general result obtained by J. P Globevnik. Theorem 2. [7] We assume that the previous conditions hold. For every ρ = (ρ ,...,ρ )∈C2(B)N in a neighborhood of r =(r ,...,r ), we set 1 N 1 N M ={ω ∈f(ζ)+B/ ∀1≤j ≤N, ρ (ω)=0}. ρ j Assumethat thepartial indices of M =M along f arenon-negative, and denoteby r κtheMaslov index ofM alongf. Then, thereexistsomeopen neighborhoods V ofr inC2(B)N, U of0inRκ+N, W off inCα(∂∆)N, andamapF :V×U →Cα(∂∆)N C C of class C1 such that • F(r,0)=f; • for all (ρ,t) ∈ V ×U, the map ζ 7→ F(ρ,t)(ζ)−f(ζ) is the boundary of a holomorphic disc glued to M ; ρ • there exists η > 0 such that ∀ρ ∈ V,∀t ,t ∈ U, ||F(ρ,t )−F(ρ,t )|| ≥ 1 2 1 2 η|t −t |, in particular F(ρ,t )6=F(ρ,t ) for t 6=t ; 1 2 1 2 1 2 • if f˙ ∈ W is the boundary of a holomorphic disc glued to M , then there ρ exists t∈U such that f˙=F(ρ,t). When the partial indices are only supposed to be larger than -1, the result has been extended by Y.-G. Oh [9]. Let h be a non-constant stationary disc centered at p = (p ,0,...,0) ∈/ Q and 0 gluedtoQ. Accordingto(6),w =h′ (0)6=0,whichallowsustoassume(tw¯A) 6=0. α n We denote by P(N∗Q) the projectivization with respect to the n−th coordinate. Let h∗ be a regular lift of h, and h∗ h∗ hˆ∗ = 0,..., n−1 h∗ h∗ (cid:18) n n (cid:19) be its projectivization. Then f = (h,hˆ∗) is the boundary of a disc glued to |∂∆ P(N∗Q), and (7) twAw 1+aζ ζw ζw a¯−ζ (tw¯A) (tw¯A) 1 n 1 n−1 f(ζ)= +iy , ,..., , , ,..., 1−|a|21−aζ 0 1−aζ 1−aζ 2(tw¯A) (tw¯A) (tw¯A) (cid:18) n n n (cid:19) 8 LEÁBLANC-CENTI in view of Remark 2. In particular, f is of class C∞ up to the boundary, and does not depend on the choice of the regular lift h∗. Wewanttoapplythe theoremofGlobevniktoM =P(N∗Q)alongtheholomor- phic disc f. Note that T(ζ)=T (P(N∗Q)) is totally real in view of Proposition f(ζ) 1. The difficulty is to compute the partialindices andthe Maslov index of P(N∗Q) along f. For the convenience of the reader, we prove the two following lemmas in the Appendices. Lemma 1. The partial indices of P(N∗Q) along f are non-negative. Lemma 2. The Maslov index κ= 2n κ of P(N∗Q) along f is equal to 2n+2. j=0 j Hence we get: P Proposition 4. Theorem 2 applies to our situation, and gives that the set of the discs we consider forms a real [(2n+1)+κ]-parameter family. 2.2.2. Discs glued to a small perturbation of Q. Proposition 4 shows that the set of all holomorphic discs near f =(h,hˆ∗), glued to a small perturbation of P(N∗Q) forms a manifold of (real) dimension 4n+3. This gives us the description of the set of all holomorphic discs near h glued to a small perturbation of Q by means of the following lemma: Lemma 3. (see [8, 11]) Given t sufficiently small and ρ near r with respect to the C3-topology, F(ρ,t) is the projectivization of a regular lift of some stationary disc. Proof. We denote by s the parameter (ρ−r,t). Then, the disc F(ρ,t) is written in coordinates (hs,...,hs,Hs,...,Hs ). The holomorphic disc hs is glued to 0 n 0 n−1 Ms =π(Ms), where Ms ={ρ=0} and π is the canonical projection on the n+1 first coordinates. Hence Ms ={ρ =0} and we may set 0 ∂ρ φs :∂∆∋ζ 7→ζ 0(hs(ζ)). ∂z n For s = 0, we obtain the initial disc f and for all ζ ∈ ∂∆, φ0(ζ) = −1 (tw¯A) . 1−a¯ζ¯ n Notethatthefunction∂∆∋ζ 7→ −1 takesitsvaluesinthehalf-plane{ℜ(z)<0}. 1−a¯ζ Thus, for every sufficiently small s, the function φs also takes its values in some open half-plane which does not contains 0. This allows us to define dz ψs :ζ 7→log(φs(ζ))= . z Z[z0;φs(ζ)] Thefunctionψs isofclassCα. SetUs =−T(ℑψs)andλs =exp(Us−ℜψs),where T is the Hilbert transform. The function λs is positive valued, andwe haveon∂∆: log(λsφs)=(Us−ℜψs)+(ℜψs+iℑψs) whichextendsholomorphicallyto∆insomefunctionalwaysdenotedbyUs+iℑψs. Hence,exp(Us+iℑψs)isaholomorphicextensionofλsφsA˜ ∆thatdoesnotvanish in ∆¯. For all 0≤j ≤n−1, ∂ρ ∂ρ λs×ζ 0 ◦hs = λs×ζ 0 ◦hs ×Hs ∂z ∂z j j (cid:18) n (cid:19) is a product of functions which extend holomorphically to ∆. The function defined byh∗s :=λs∂ρ◦hsisthusaregularliftofhs. Hence,(hs,Hs)istheprojectivization ∂z of (hs,h∗s). Moreover,ζh∗s does not vanish in ∆¯. (cid:3) n STATIONARY DISCS GLUED TO A LEVI NON-DEGENERATE HYPERSURFACE 9 This proves that the map (h,h∗) 7→ (h,hˆ∗) is onto. For any stationary disc h glued to M, and h∗(1), h∗(2) two regular lifts of h: ∀ζ ∈∂∆, ∃λ(ζ)∈R∗/ h∗(2)(ζ)=λ(ζ)h∗(1)(ζ). Since ζh∗(j) is holomorphic and does not vanish in ∆¯ by hypothesis, the function n ζh∗n(2) =λextendsholomorphicallyto∆: henceλisarealnon-zeroconstant. Then ζh∗n(1) the stationary disc h has an only regular lift up to multiplication by a real non- zero constant, and two regular lifts have the same image under (h,h∗) 7→ (h,ˆh∗). Consequently, the map h7→(h,hˆ∗) is a one-to-one correspondence. Finally, we get that the set of all the stationary discs near h0 glued to a small deformation of Q form a manifold of (real) dimension 4n+3. More precisely, we have proved: Theorem 3. Let B ⊂ R2n be an open ball in a neighborhood of 0, A an n-sized non-degenerate hermitian matrix, and Q = {0 = r(z) = ℜz − tz¯Az }. Let h0 0 α α be a stationary disc glued to Q such that h0(0) = (p ,0,...,0) ∈/ Q. Then there 0 exist some open neighborhoods V of r in C3(B,R), U of 0 in R4n+3, W of h0 in Cα(∂∆)n+1, and a map H:V ×U →Cα(∂∆)n+1 of class C1 such that: C C • H(r,0)=h0; • for all (ρ,t)∈ V ×U, H(ρ,t) is the boundary of a stationary disc glued to M, where M ={ρ=0}; • if t 6=t , H(ρ,t )6=H(ρ,t ); 1 2 1 2 • if h˙ ∈ W is the boundary of a stationary disc glued to M = {ρ = 0}, with ρ∈V, then there exists t∈U such that h˙ =H(ρ,t). Note that, by Corollary 1, there exists a non-constant stationary disc h0 glued toQandsuchthath0(0)=(p ,0,...,0)ifandonlyifthefollowingconditionholds: 0 Condition ∗ The point p=(p ,0,...,0) verifies p∈/ Q. Moreover, ℜp >0 (resp. 0 0 ℜp <0) if A is positive definite (resp. negative definite). 0 Corollary 2. For ρ near r with respect to the C3-topology, the map Φ−1 defined in (5) is always a local C1-diffeomorphism from R×Cn ×(Cn \{0})×∆ to the set Mρ of all non-constant stationary discs glued to M ={ρ=0}. Proof. We use the notations of Theorem 3 and Proposition 3. For all ρ ∈ V, we define the map Θ : U → R×Cn×(Cn\{0})×∆ ρ t 7→ (y(ρ,t),v(ρ,t),w(ρ,t),a(ρ,t)) 0 where y(ρ,t),v(ρ,t),w(ρ,t),a(ρ,t) arethe parametersdefining the disc H(ρ,t). We can 0 assumethatanydisch=H(ρ,t)verifies(th¯′ (0)A) 6=0. ThemapΘ isofclassC1. α n ρ Moreover,themapµ:(ρ,t)7→d(Θ ) iscontinuousfromV×U totheBanachspace ρ t L (R4n+3)ofcontinuouslinearmaps. Sinceµ(r,0)isinvertible,wecanassumethat c the differential of Θ is invertible at any point. This gives the conclusionby means ρ of the inverse function theorem. (cid:3) 2.3. Discs centered at p. We can now give two parametrizations of the discs centeredatsomefixedpointandgluedtoasmallperturbationofQ. Webeginwith studying discs glued to Q. 10 LEÁBLANC-CENTI Proposition 5. Pick p = (p ,0,...,0) verifying Condition ∗. Then the map that 0 associates to any non-constant stationary disc h, glued to Q and centered at p, the point h(1) (resp. the vector h′(0)), is a local diffeomorphism on its image. Proof. The map h 7→ h(1) has been studied yet in Corollary 1. For h 7→ h′(0), according to Proposition 3 and Remark 2, it suffices to consider the map tw¯Aw Ψ:(Cn\{0})×∆∋(w,a)7→ 2a ,w . 1−|a|2 (cid:18) (cid:19) If Ψ(w,a)=Ψ(w˙,a˙), then w=w˙ and a a˙ |a| = |a˙| |a|=|a˙| = ⇔ 1−|a|2 1−|a˙|2 ⇔ 1−|a|2 1−|a˙|2 ( Arga=Arga˙ (cid:26) Arga=Arga˙. Hence Ψ is injective and induces a homeomorphism on its image. Moreover, 2atw¯′Aw+tw¯Aw′ + 2 tw¯Aw (a′+a2a¯′) dΨ (w′,a′)= 1−|a|2 (1−|a|2)2 , (w,a) w′ + 0 ! thus dΨ is injective at every point. (cid:3) By means of arguments similar to those of the proof of Corollary 2, one gets Theorem 1. Remark 4. More precisely, the map h7→(ℑh (1),h (1)) is a local diffeomorphism 0 α on R×Cn. 3. Uniqueness property Let M be a small perturbation of the hyperquadric Q. As soon as z ∈Q is met by some stationary disc, Theorem 1 provides, in a neighbourhood of z, a foliation of M by the boundaries of stationary discs. The following lemma determines such points z: Lemma 4. Assume that the conditions of Theorem 1 hold. We denote by δρ the map defined on M(ρ) by δρ(h)=h(1). p Then the image of the map δr is exactly {z ∈Q/ ℜz ℜp >0}. In particular, if A 0 0 is positive definite or negative definite, then every z ∈Q\{0} may be written under the form z =h(1) for some h∈M(r). p Proof. Let h be a stationary disc glued to Q and centered at p= tw¯Aw +iy ,0 . 1−|a|2 0 Thus, for z ∈Q: (cid:16) (cid:17) tw¯Aw 1+a 1 w =(1−a)z α z=h(1)=(cid:18)1−|a|2 1−a +iy0,w1−a(cid:19)⇔(cid:26)z0 6=−ℜp0+iℑp0 and a= zz00+−pp¯00 . It admits a solution if and only if z 6=−p¯ and 0 0 z −p (ℜz −ℜp )2+(ℑz −ℑp )2 0 0 0 0 0 0 <1⇔ <1⇔ℜz ℜp >0, z +p¯ (ℜz +ℜp )2+(ℑz −ℑp )2 0 0 (cid:12) 0 0(cid:12) 0 0 0 0 (cid:12) (cid:12) which co(cid:12)ncludes(cid:12)the proof. (cid:3) (cid:12) (cid:12)