GEOMETRY AND ANALYSIS OF CAUCHY-RIEMANN MANIFOLDS
by
WONG WAI KEUNG
A Thesis Submitted to the Graduate School of The Chinese University of Hong Kong (Division of Mathematics) in Partial Fulfilment of the Requirement for the Degree of Master of Philosophy (M. Phil.)
HONG KONG June, 1998 f^ii^k^^/#^^v � |j( 1 6 Jbl 1399 ]i)) 、家 ”_ERSITY”/M ‘^aiBL.RY SYSTEMy^^ �>>�^J^!^^：^^^^^ ^li^^ ACKNOWLEDGEMENT
I would like to express my deepest gratitude to Prof. H. S. Luk for his advice and kind supervision. I also wish to thank Orieta Wong for her tolerance and support during my study.
Wong Wai Keung
Department of Mathematics The Chinese University of Hong Kong June, 1998
w‘. ‘ • . 摘要
在這篇論文中，我們討論柯西一黎曼流形（簡稱0尺流形）的幾何及分析理
論。其實，0尺流形是複流形中實子流形的抽象模型o我們首先定義抽象的CR 流形（這並不一定爲實超曲面類型），並討論嵌入€尺流形，我們會就一般的嵌
入CR流形提出一種正規形式。接着，我們會參照Tanaka的工作，介紹強僞凸 的0尺流形的微分幾何硏究。這硏究以聯絡和曲率組成一套極佳的形式體系，但 這體系取決於如何選擇一個基本的一次型。雖然如此，我們可以應用這些理論來
硏究CR流形M上任何全純向量叢E相伴複形{ ^ ( M, E )》£}的調和理論。這譎 和理論是半橢圓算子理論的典型例子。當我們將此用在全純切叢和它的外積時’
我們便得到正切的柯西一黎曼複形{0’9(河),^M}，並復得Kohn的調和理論°
我們在這論文中討論的另一個分析課題是在一般的嵌入CR流形上CR函數 的全純擴張性。我們會介紹Boggess和Polking在一般的嵌入CR流形上的一個 全純擴張定理，其中利用了所謂“分析圓盤技術”，並運用Baouendi和Treves 的一個近似定理。 ABSTRACT
In this thesis, we discuss the geometry and analysis of CR manifolds, which are abstract models of real submanifolds in complex manifolds. We first give the definition of abstract CR manifolds (not necessarily of the real hypersurface type) and discuss embedded CR manifolds, presenting a normal form for a generic embedded CR manifold. Then we present a differential geometric study of strongly pseudo-convex CR manifolds, following the work of Tanaka. There is a very nice formalism of connection and curvature, which, however, depends on the choice of a basic one-form. Nevertheless, the geometry can be applied to study the harmonic theory of the associated complex {C%M, E), \} of any holomorphic vector bundle E over M. This harmonic theory is a prototype of the theory of subelliptic operators. When we specialize to the holomorphic tangent bundle and its exterior products, we obtain the tangential Cauchy-Riemann complex {C^'%M), d^} and recover Kohn's harmonic theory.
Another analytic aspect which we study is the holomorphic extension of CR functions on generic embedded CR manifolds. We present Boggess and Polking's holomorphic extension theorem for generic embedded CR manifolds using the technique of analytic discs. For that purpose, we also prove an approximation theorem of Baouendi and Treves. Contents Abstract
Litroduction 1
1 CR Manifolds 3 1.1 Abstract CR manifolds 3 1.2 Embedded CR manifolds 4 1.3 A normal form for generic embedded CR manifolds 9 2 Differential Geometry of Strongly Pseudo-convex Manifolds 18 2.1 Holomorphic vector bundles 18 2.2 The cohomology groups H^M, E) 20 2.3 The spectral sequence {ErP,q(M)} 23 2.4 The Levi form 31 2.5 Strongly pseudo-convex manifolds 37 2.6 Strongly pseudo-convex real hypersurfaces 40 2.7 Canonical affine connections 44 2.8 Green's Theorem 51 2.9 Canonical connections in holomorphic vector bundles 53
3 The Harmonic Theory 59 3.1 The fundamental operators 59 3.2 The fundamental inequalities 65 3.3 Kohn's harmonic theory 67 3.4 The harmonic theory and the duality 71
4 The Holomorphic Extension of CR Functions 76 4.1 Approximation theorem 76 4.2 The technique of analytic discs 81 4.3 Holomorphic extension 95
Bibliography 101
i Introduction
Cauchy-Riemann manifolds (in short CR manifolds) are abstract models of real submanifolds in complex manifolds, in particular, the boundary of domains or complex subvarieties in C". As such, they play an important role in several complex variables. In this thesis, we discuss the geometry and analysis of CR manifolds related to the following two areas: (1) the tangential Cauchy-Riemann complexes, which is related to the solutions of a-Neumann problem and (2) the holomorphic extension of CR functions, which is related to the solutions to the homogeneous tangential Cauchy-Riemann equations. It was Kohn [K] who started the work in the first area in the 1960s. He used a Hilbert space approach to construct solutions to the tangential Cauchy-Riemann complex on the boundary of a strictly pseudo-convex domain. Later, Tanaka [Ta] made a differential geometric study on the cohomology groups of the tangential Cauchy-Riemann complex, which is based on the harmonic theory developed by Kohn. The second area was started by Hans Lewy CL] in the 1950s. He showed that under certain convexity assumptions on a real hypersurface in C", CR functions locally extend to holomorphic functions. Many refinements to the extension theorem have been made over these years so that weaker convexity assumption is needed. One of the significant approaches is the technique of analytic discs applied to the CR manifolds of higher codimension, which is originally due to Lewy ¢.] and Bishop DBi].
In this thesis, we are going to investigate some of the work of Tanaka on the study of harmonic theory for the tangential Cauchy-Riemann complexes. Moreover, we employ the technique of analytic discs to prove a holomorphic extension theorem for CR functions due to Boggess and PoUdng pP]. The following is the arrangement of this thesis.
I Chapter 1 covers general information about CR manifolds. We give the definitions of abstract CR manifolds (not necessarily of hypersurface type) and discuss embedded CR manifolds. In addition, we present a normal form that gives a convenient description of a generic embedded CR manifolds in local coordinates. In chapter 2, we specilize to a strongly pseudo-convex CR manifold M. We introduce the notion of a holomorphic vector bundle E over M and define the associated complex {C\M, E), \} where \ is the Cauchy-Riemann operator of E. Then we introduce a 2 Introduction filtration of the de Rliam complex of M and let E^'^(M) be the associated spectral sequence. The cohomology groups W'^ are the Kohn-Rossi cohomology groups. We also discuss the canonical affine connections V of M and the canonical connections D of a holomorphic vector bundle E over M.
Chapter 3 is devoted to the analysis for the complex {C%M, E), ^}. We first introduce the Laplacian operator U^ in the harmonic theory, and express it in terms of the covariant differentiation which is induced from the canonical connections V and D defined in the last chapter. We then prove subellipticity for Dg and obtain Kohn's harmonic theory. We apply the harmonic theory to the tangential Cauchy-Riemann complex {^'P'^(M), dj^} where d^ is the tangential Cauchy-Riemann operator of M. Finally, we prove a duality theorem on the cohomology groups IP,q.
In Chapter 4，we start with an approximation theorem of Baouendi and Treves D5T]. We follow Boggess and PoUdng's application of the technique of analytic discs in proving a holomorphic extension theorem on generic embedded CR manifolds. The proof requires an analysis of the solution of a nonlinear integral equation involving the Hilbert transform. Finally, we observe that the technique of analytic discs gives an easy proof of Hans Lewy's holomorphic extension theorem for hypersurfaces. Chapter 1 CR Manifolds
We first introduce the definition of abstract CR manifolds. After fixing the notation and definitions, we shall give some examples of CR manifolds. Then we discuss the generic embedded CR manifolds Po][T]. 1.1 Abstract CR manifolds
For any real manifold M, we denote the real tangent and cotangent bundles of M by T(M) and T(M) * respectively. Then the complexified tangent bundle of M is denoted by CT(M) and CT*(M) is called the complexified cotangent bundle. For any p E M, CTp(M)* is the dual space of CTp(M) by the pairing
DEFEVITION 1.1.1. Let M be a real manifold and H(M) be a subbundle of T(M). A smooth vector bundle isomorphism J: H(M) — H(M) is called a complex structure map if J o J = -I where I ： H(M) — H(M) is the identity map.
Clearly, if dim^H(M) = n, n is an even number since (detJ)^ = (-1)". J is extended as a complex linear map, called J again, on CH(M) by defining J(X0X) = XJ(X) where X G H(M) and X G C. The eigenvalues of J are +i and -i with corresponding eigenspaces denoted by S and S. Thus, CH(M) is uniquely decomposed into two complex subbundles with CH(M) = S©S. It is easily checked that S = {X^ - iJX^ : X^ E H(M), 1 < k < n}. Likewise, S = {X, + iJX, : X,GH(M), 1 EXAMPLES 1.1.3. Trivially, an integrable ahnost complex structure of a real manifold M is a CR structure. In this case, CT(M) = S © S’ S 门 S = {0} and [T(S), T{S)] C T(S). 1.2 Embedded CR manifolds Let us review the complex structure of C". Let (z” ...,zJ be the local coordinates for C" where Zj = Xj +iyj, Xj and y� ME, 1 < j < n. The complex structure forCT(C") is defined by J(~^) = — and J( — ) = --^ where 1 < j < n. oXj dYj ayj dx. • Define the vector fields • � ^ � d 1 d . d , d 1 d . d 1 ,., —=一 —-1 — and - = 一 +1 1 < 1 < n dz, 2 dx, dy, dz, 2 ax. ay. ‘ -J-^- J J J j j J 、 乂 V - The corresponding bundles will be denoted by T^-^(C") and T^-^(C"). They are called the holomorphic tangent bundle and the anti-holomorphic tangent bundle respectively. Clearly, TO,i(cn) = T1.0(C"), Ti,0(cn)门 T0’i(cn) = {0}, CT(C) = Ti，°(cn) �T°’i(cn)， and [r(Ti,o(eo), r(T"(cn))] C r(Ti'o(cn)). Chapter I: CR Manifolds 5 Let M be a smooth real submanifold of C" of real codimension d. That is, diniRM : 2n -d. For each p E M, we define a subspace Hp'° (M) of CTp(M) by H^'(M) = CTp(M)门 i;i'Vcn) and HpO'i(M) = CTp(M)门 TpO,i(Cn). Clearly, H” (M) = H;'u(M). DEFEVITION 1.2.1. Hpi,o (M) is called the holomorphic tangent space to M at any point p G M. For any locally defined smooth submanifold of C"，the holomorphic tangent space can be identified by the following proposition. PROPOSITION 1.2.2. Let M be a smooth submanifold of C" locally defined by a set of smooth real-valued functions {p^,…，pJ with dp^ A …K dyOd 5^ 0 near a point p G M. Suppose X = ^ a A G T^'^e) and Y = T p.丄 G T^'^C"). j = 1 d z. j = 1 J d Zj (a) X E Hpi,o(M) ifand only if (b) Y e HpO’i (M) if and only if PROOF. We give the proof of part (a) only and part (b) is similar. The basic theory of differential geometry tells us that for any vector X G Tp'°(C"), • Chapter I: CR Manifolds 12 The above results imply that dim^ Hp'° (M) = n - k where k is the number oflinearly independent elements of { dp^，…，dp^}. PROPOSITION 1.2.3.[T] Let M be a smooth real submanifold of C" with dim^M = 2n-d, where d is the real codimension of M. Then max(0,n-d) < dinXcHp'^M) < n-_^. 2 PROOF. Clearly, dim^Hp'-'CM) < l-dun^CT^(M) = n-^. 2 2 Furthermore, U'/ (M) C CTp(C"),this gives dimcCTp(M) + dimcTpi’。（Cn) - dim^Hp^'^M) < dim^CTpCC"). Therefore, dim^Hp'° (M) > 2n - d + n - 2n = n - d. The first inequality is then established. > • Note that dim^Hp'^ (M) may be different for different point p G M. DEFEVmON 1.2.4. Suppose that M is a real submanifold of C". M is called an embedded CR manifold if dinicHpi'o(M) is constant for any p E M. EXAMPLE 1.2.5. Here are some examples of embedded CR manifoids: (a) The real hypersurfaces of C" are clearly the embedded CR manifolds. Suppose M is a real hypersurface with real codimension d = 1. Then we have dimcHpi,o(M) = n - 1 for any p G M. (b) A submanifold M of C" is called a totally real submanifold if Hp'^M) = {0} for each p G M. In this case, dinicHp'^ (M) is constant for any p G M. Chapter I: CR Manifolds 7 In the rest of this chapter, suppose that M is an embedded CR manifold. Then H"(M) = U Hpi,�(M)andHO,i(M =) U Hp''(M) are subbundles of CT(M). In peM p e M fact, if M is locally defined by { p^，•..，p^} near p G M，then we can choose a set of k linearly independent forms { dp. , ...，dp^J from { 5p^, ...，dp^} such that there are n-k smooth linearly independent vector fields {X^} with The pair (H(M), J) is called the real expression of Hi’�（M). DEFINITION 1.2.7. Let Gp(M) = Tp(M)/Hp(M). This quotient space is called the totally real part of Tp(M). DEFMTION 1.2.8. dim^ Hp(M) is called the CR dimension of M at p. dinij^Gp(M) is called the CR codimension of M at p. Clearly, by Proposition 1.2.3.,we have 0 < dim^Gp(M) < d. Chapter I: CR Manifolds 8 In the following we shall identify Gp (M) with the orthogonal complement of H (M) in T (M) with respect to the Euclidean inner product. Clearly, J{G^ (M)} is r r y orthogonal to Hp (M) and J{Gp (M)}门 Gp (M) = {0}. Hence, we have Tp(M) =Hp(M)0Gp(M). DEFEVITION 1.2.9. An embedded CR manifold M is said to be generic if the CR dimension of M is minimal, i.e., if dinij^Hp(M) = 2n-2d. Equivalently, M is generic if the CR codimension of M is minimal (=d). PROPOSITION 1.2.10. Suppose that M is an embedded CR manifold locally defined by a set of smooth real-valued functions { p^, ..., p^} with dp^ A ... A dp^ ？^ 0 near a point p E M. Then the following statements are equivalent: (a) M is generic. (b) dpiA ... Aapd 卢 0. �. (c) Tp(Cn) = Tp(M) 0J{Gp(M)),i.e. Tp(Cn) = Hp(M) e Gp(M) © J{Gp(M)}. PROOF. If dpi/K ... AaPd 5^ 0，then dinicHp''(M) = n-d = dimcHp'^(M). Hence, din^Hp (M) = 2n - 2d. The decomposition in (c) follows immediately. • EXAMPLE 1.2.11. (a) A real hypersurface M of C" is generic since the CR codimension of M is 1. (b) In the above example, J{Gp (M)} is orthogonal to Gp (M). In general, this may not be true. Let M = {(Zi, Z2) E C2 ： Imzi = Rez^ and Imz^ = 0}. At (0, 0), To(M) is generated by {各，^ + 各}. d Xj 3 y^ d x^ Chapter I: CR Manifolds 15 The complex structure is defined by J(/_)=/-,J(^./_) = -^./.. ^ Xi d y, d y, d x^ d x^ d y^ 3 ^ p^ Note that J( —^ Xi ) is not orthogonal to _d y^ +_^d x^. They are not contained in To(M). Hence, H。（M) = T^(M)门 J{T。（M)} = {0).Misnon-generictotally real embedded CR manifold. 1.3 A normal form for generic embedded CR manifolds In this section, we construct a normal form for a generic embedded CR manifold. LEMMA 1.3.1. Suppose M is a generic embedded CR manifold with real dimension : 2n -d, 1 < d < n. Let Po be a point in M. Then there is a non-singular, complex affine linear map A : C" — C", an open neighbourhood U of Po, and a smooth function h : R^ x C" _ ^ — R^ with h(0) = 0 and Dh(0) 二 0 such that A{Mf|U} = {(z,w) G C^ X Cn-d ： imz : h(Rez,w)}. PROOF. Take Po to be the origin. It suffices to find a linear map A that takesT�（M) to {Im z = 0}. Since M is generic, dim^Go(M) : d and di^Ho(M) = 2n-2d. Let {�”…， vbJe an orthonormal basis for G^(U). Hence, {Jv^,…，JvJ is an orthonormal basis for jq(M) and{�d+i，J�d+i，...，々n，J�n}isanorthonormalbasisforH 。（By PropositioM). n 1.2.10.,{V,, Jv^, ..., v^, Jvj is a basis for T^(C") = T^(R^^). Let z = X +iy G C^ and w = u +iv E C""^ where x, y G R^ and u, v G M"-^ Define a real linear map A : R^" — R^" by 各 A(v.J) = Ad Xj, A(Jv)=j d y. for 1 AoJ(vp = A ^j = J(^Xj) =JoA(Vj) for 1 THEOREM 1.3.2. Suppose M is a real analytic, generic embedded CR manifold with dim„M = 2n-d K ， 1 < d < n. Suppose p^ G M. There is a neighbourhood U of Po in C", a biholomorphism cp ： U^<;^(U) E C", and a real analytic function h: R^ x C""^^M^ with h(0) = 0 and Dh(0) = 0 such that (1) WMf|U} = {(z,w) G <^(U) C Cd X Cn-d ： Imz=h(Rez,w)),and ^«卜丨到h …“丨+丨引h (2) ^_^(0) = 0， =^(0) = 0, for all multi-indices a and 3, dx"dw^ ax"5w^ where x = Rez E R^ PROOF. We first choose the coordinates (z, w) G C^ x C"~^ where z = x +iy, X，y G 1^，so that Po is the origin and M = {(X +iy, w) G Cd x C""^ : y =h(x, w)} with h(0) = 0，Dh(0) = 0. Since M is real analytic, the function h is also real analytic and its Taylor expansion about Chapter I: CR Manifolds 11 the origin is given by h(x,w,w) = Y^ a^^^x"w^w^ a,/3,7 1 ^l"hl/3hl7l h where a . = • (0). ah al|3lyl ax"aw^aw^ Note that h depends on w in this expansion. Now, we replace w by an independent variable 7} G Cn-d and extend the function h naturally to h: C^ X C""^ X C""^^C^ by fi(z,w,ry) = Y^ \0,z«wV a./3.7 which converges in the set { | z | < 6，| w | < 8， | rj | < 8} for some 8 > 0. Since h(0) = 0 and Dh(0) = 0，h(0) = 0 and Dh(0) = 0. By implicit ftinction theorem, there is a biholomorphic map g : C^ x C" “ ^ — C^ defined near the origin such that g(z + ih(z,w,0), w) = z for I z I，I w 丨 < b'，b' > 0 and b' < 8. Define A for (z,w) = (p{z, w) E M. Since ^ x(x,w,w) is real analytic near the origin, it can be expanded in x, w, and w. We replace w by rj E C""^ and x by z E C^ in x(x,w,w) and we have a holomorphic map x: C^ x C""^ x C"'^^C^ near the origin. A Hence, a new function h is defined by h(z,w,^) = h(x(z,w,^),w,^) ~ -h(g(x(z,w,r))+ih(x(z,w,^),w,^),w),w,0)...... (4) which is holomorphic for (z,w,w) in a neighbourhood of the origin. Recall that the holomorphic map g is defined near the origin by g(z + ih(z，w ,0), w) = z. Replace z by x(z, w, 0) and w by w，we have g(x(z,w,0) +ih(x(z,w,0),w,0), w) = x(z,w,0). Put this into (4)，the result is thus obtained. Bi Chapter 1: CR ManifoUh ;5 The graphing function h defmed is real analytic and it is expressed in a series h(x,w) = V a . x"w^w^ about the origin. The terms with coefficients either a ^ or a.^.7 ",。’？ a«,i8,o ^e called the pure terms. Hence, the second condition in the theorem restricts the choice of coordinates such that the graphing function of M has no pure terms in the Taylor expansion. We shall see that a C^ (k > 2) manifold M agrees with a real analytic embedded CR manifold to order k and therefore, we have a C^ version of Theorem 1.3.2. Suppose M = {(z,w) E C^ x C"^ : Imz=h(Rez,w)} wherehisofclass C^ for (k > 2). Then the kth order Talyor expansion of h about the origin is given by h(x,w) = hj(x, w, w) +r(x, w) where h^ is a polynomial of degree k and r is the remainder of expansion satisfying ol"IH)8hl7l “一 a (3 | + | 7 -~d x«, d w^ d w^ r(0 ) = 0 for I | + | ‘ | < k. Hence, M agrees with N = {Imz = hj(x, w, w)} to order k at the origin. Theorem 1.3.2. is then applied to N and we have the following C^ version. THEOREM 1.3.3. Suppose M is a generic embedded CR manifold of class C^ (k > 2) with dim^(M) = 2n -d, 1 < d < n. Suppose p� M€ . There is a neighbourhood U of po in C", a biholomoq)hism cp :U — ^(U) G C", and a function h : M^ x C""^^R^ of classC^ with h(0) = 0 and Dh(0) = 0 such that (1) HMflU} = {(z,w)GAU)C CdxCn-d ： toz=h(Rez,w)),and ai"i*i^i h a""i*i^i h ， 0，for (2) 4~^(0d x« d w^ ) = 0 1——^(0ax"5w^ ) = I OL I + I 0 I < k, where x =Rez G R^. Chapter I: CR Manifolds 14 n n 0 0 Recall that {-_，…’ } and {_t_,…，_i_} are the bases for � � -d awi aw„./ H^'o(M) and Ho'^ (M) respectively. We shall extend these vectors to vector fields which are the canonical local bases for H^'^(M) and H°'^ (M). PROPOSITION 1.3.4. Suppose M = {(z,w) G C^ X C""^ : Imz=h(Rez,w)} where h is of class C^ for (k > 2) with h(0) = 0 and Dh(0) =0. Abasis {X^, ...,X„.J for H^-^M) neartheorigin is defined by XJj =Jd ~w. -2i,¾£ \k4^d Wj d |z, forl PROOF. LetX. = ^ -EA,.^Gr'0(M)forl Note that since Dh(0) = 0，X. |j^ =各 and X. |^^ = _L，for 1 < j < n -d. 0 Wj d Wj We have shown that a smooth generic embedded CR manifold has a locally defined graphing function h with no pure terms in its Taylor expansion up to a given order. In particular, the quadratic terms in the expansion of h has no x-terms. We introduce the quadric submanifold of C" which can be used to approximate any generic embedded CR manifold. DEFEVITION 1.3.5. A map q : C^ x C^ — C^ is a quadraticform if (a) q is bilinear over C, (b) q is symmetric, and (c) q(z, w) = q(Y,w) forz,w E C™. DEFEVITION 1.3.6. A submanifold M of C" is called a quadric submanifold of C" if it is defined by M = {(X + iy,w) G Cd x C""^ : y : q(w,w)} where q : C"'^ x C"'^ — C^ is a quadratic form. EXAMPLE 1.3.7. The Heisenberg group M is one of the most important examples of a quadric submanifold. It is a real hypersurface in C" defined by M = {(z,w) E C X Cn-i ： Imz = | w�}. Define the operation o : M X M — M by (Zi,Wi) o (Z2,W2) = (Zi +Z2+2iWi.K, Wj +W2) for (Zi, Wi), (¾, W2) G C X Cn-i. SinceIm{Z1+Z2+2iW1.W2} = IniZj + ImZ2 + 2Wj -w^ = I w^ + W2 I^, M X M is closed under 0 . Clearly the operation is associative. The origin (0，0) G C x C""^ is the identity, (z,w)"^ = (-z +2i | w p, -w) E M is the inverse of (z, w). Therefore, the operation 0 ： M x M — M defines a group structure on M x M. It implies that the Chapter I: CR Manifolds 22 operation o ： M x M — M defines a group structure on M x M. It implies that the Heisenberg group is a Lie group. For Po = (ZQ,WQ> G M, define a map Gp� M: — M by Gp�(z， w=) (z,w) o (Zo,Wo). Gp� ai ssmooth function in both (z, w) and p� If. we fix p^ G M，Gp� thise restriction of a holomorphic map since the inner product is complex linear as a function of w. Note that the map (z,w) >^ Gp。（z，w) sends the origin (0’ 0) to p� Therefore. ，（Gp)*(0) is a complex linear map from CTo(M) to CTp (M). By Proposition 1.3.4., the generators for H^'^(M) near the origin is given by Xj =各 ^ 2iw.^, l We conclude this chapter by stating a normal form for the quadric surfaces of real codimension 2 in C^. Let (z^,z2,w^, w^) be the coordinates for C^, wherez^ = x^ + iy^ and Z2 = x^ + iy2. A quadric of real codimension 2 in C^ has the form M = {(Z1,Z2,W1,W2) G 0:y^ =qi(w’—’y2=q2(w，W，w E C^} where q^ and q^ are scalar-valued quadratic forms in C^. q^ and q^ are said to be linearly independent over R if the restriction of q^ and q^ to the set {(w，w): w E C^} are linearly independent over M in the usual sense. Chapter I: CR Manifolds 23 THEOREM 1.3.8. Suppose that M is a quadric submanifold of C^ with real codimension 2 as defined above. (a) If q^ and q^ are linearly dependent over M, then by a suitable non-singular complex linear transformation, M is of the form M = {(Z1,Z2,W1,W2) G C4:y! =q/w,w),y2=O,wG C2} where q^ is a scalar-valued quadratic form. (b) If q^ and q�ar linearle y independent over 1, then by a suitable non-singular complex linear transformation, M is of one of the following forms: (1) M = {(Zi’Z2’Wi’W2)e c^: Yi = |Wi p,y2= |W2|2} (2) M = {(Zi，、，Wi，W2) G C4:yi = I Wi |2，y2=Re(Wi�2)} (3) M = {(Z1,Z2,W1,W2) E C^:y1=Re(W1W2),y2=Im(W1W2)}. Chapter 2 Differential Geometry of Strongly Pseudo-convex Manifolds In this chapter, we shall define holomorphic vector bundles and the associated Cauchy- Riemann complexes. A relevant filtration of the de Rham complex of M will be introduced and the associated spectral sequence [GH] PM] will be studied. We also define the Levi form for a CR manifold M. The Levi form measures how far the subbundle CH(M) fails to be closed. We shall introduce the notion of a strongly pseudo-convex manifolds. The canonical affine connection of M and the canonical connections of a holomorphic vector bundle E over M are discussed [Ta]. • 2.1 Holomorphic vector bundles Let M be a CR manifold with a CR structure S and CF(M) denote the space of all smooth complex-valued functions defined on M. DEFmXION 2.1.1. A complex vector bundle E over M is said to be holomorphic if it is equipped with a differential operator d^ : F(E) — F(E0S *) which satisfies the following conditions: (1) X(fu) = Xf-u + f.Xu; (2) [X, Y]u = XYu — YXu, ‘ where u G r(E),fG CF(M), X, Y E r(S) and Zu denotes (4u)(Z) for Z G r(S). The operator d^ is called the Cauchy-Riemann operator of the holomorphic vector bundle E. If u E r(E) satisfies 3gU = 0, then u is called a holomorphic cross section of E. EXAMPLE 2.1.2. Suppose E and F are two holomorphic vector bundles over M. For u E F(E),v E r(E) and X E F(S), the tensor product E ® F is also a holomorphic vector bundle if the Cauchy- Riemann operator 3gg,p(u0v)(X) = X(u0v) satisfies the rule: X(u0v) = (Xu)®v + u0(Xv). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 19 ‘ This differential operator satisfies (1) and (2) in 2.1.1. Indeed, for f G CF(M), X(fu0v) = (Xf)u0v + f[(Xu)0v + u0(Xu)] = (Xf)u0v + fX(u®v). Also, [X，Y](u0v)=XYu0v + u0XYv - YXu0v - u®YXv = XY(u0v) - YX(u0v). Suppose t(M) is a quotient bundle of a CR manifold M defined by t(M) = CT(M)/S. Let w : CT(M) — t(M) be a natural projection. For any u G r(t(M)), we can find Z E r(CT(M)) such that u = w(Z). For any X E r(S)，define a differential operator at(M) • r(t(M)) — r(t(M)0s*) by (\(M)u)(X) = Xu = &([X, Z]), where Z G r(CT(M)) and u = w(Z). The following proposition shows that t(M) is a holomorphic vector bundle with respect to the Cauchy-Riemann operator 3细). PROPOSITION 2.1.3. (1) 3f(M) is well-defined, i.e. independent of the above choice of Z E F(CT(M)). (2) �M is) a Cauchy-Riemann operator. PROOF. To establish (1), let u = co(Z) = w(Z+Y) where Y G T(S). Since [X, Y] C r(S), 0)[X, Y] = 0, we have w[X, Z + Y] = (i[X,Z] + w[X, Y] = ^[X, Z]. Let fu = w(fZ). X(fu) = o)[X,fZ] = dj((Xf)Z)+co(f[X,Z]) = (Xf)u + fXu. Moreover, [X, Y]u 二 (iIK[[X, Y]，Z]) = XYu - YXu by using the Jacobi identity. Hence, both the conditions in 2.1.1. are satisfied. Therefore 5卞(^) is a Cauchy-Riemann operator. • Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 20 ‘ DEFCSflTION 2.1.4. The holomorphic vector bundle t(M) is called the holomorphic tangent bundle of M. 2.2 The cohomology groups H^OM, E) Let E be a holomorphic vector bundle over M. We denote the space E ® A^ S * of all E-valued q-forms on S * by C^(M,E) and the space of all smooth cross sections of 0(M,E) by C^(M, E). DEFEVITION 2.2.1. Define the differential operator ^ : 0(M, E) — ^+^(M, E) by c^p)(Xi,...,5^q+i) = ^ (-iy+iXi(p(Xi，...’Xi，...，3^ )) + i=i A A 5^ (-l)i+j A for In particular, ^ = \ as defined in2.1.1. If cp G C\M, E), and XpX: G r(S), then (4^)(x,,3q) = x,(^(3q))-iq(^(x,))-^([x,,^]). The definition is analogous to the formula for the usual exterior differentiation. PROPOSITION 2.2.2. For any cp G ^(M, E), 'dl PROOF. Clearly ~d\(p is linear over C. Also, it is linear over smooth complex-valued functions fE CF(M). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 21 ‘ Suppose 5^1 = x! (-l)i“3^i(Wi，...，Xir.，i^q+i)) and i = l A A h = X^ (-l)"J<;c»([Xi’Xj]，Xi’...，Xp...，Xj，...，Xq+i). 1 ^ i A fEi + E (-irUX,f)^(X,,...,X,,...,X^^^) and 1 )(Xj, ... , fXk,…’ Xq + 1) = f(� The differential operator d^ satisfies a product rule which is analogous to the product rule for the exterior derivative. � PROPOSITION 2.2.4. For PROPOSITION 2.2.5. The sequence of operators dl : ^(M, E) — 0+i(M，E) is a complex. PROOF. It suffices to show that 赶+丄 o Og = 0 by induction on q. For q = 0，let f G <^(M, E) = CF(M) and X” X, G F(S), Cd'Mf))(X,X) = (¾(^))(^,,¾) =X,(^f(X,))-3q(V(Xi))-^f([X,,^]) =X,(^f)-^(XJ)-[X,,X,]f = 0. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 22 ‘ Let -¥沪([3^1’1]，&)+究“[3^1，3^3]，3^)-巧^([&，3^3]，又1) =xj3q(^(X3))-X3(^(3q))-^([3q,X3])] -X,[X,(^(X3))-X3(^(X,))-^([X,,X3])] ^x3[x,(^(3q))-^(^(x,))-^([x^,iq])] -[Xi，t](p(3^3))+3^3(p([3^i,3^2]))-p([[Xp《]，3^3]) + [3^i,l](W^))-3^(W[3^i,i^3])) + p([[^i,3^3],&]) -[&，3^3](p(Xi))+Xi(p([&，3^3]))-vK[[3^2，^3]，Xi]) =0 Now, assume the proposition is true for any (p E C^(M, E). Let rj E (?^+i(M，E). Since?/ can be expressed as the sum of the terms of the form 6 八(p，where 6 G C\M, E) and =dld'jA =0 • DEFEVITION 2.2.6. The cohomology groups of the Cauchy-Riemann complexes defined in 2.2.5. are denoted by H^M, E). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 23 2.3 The spectral sequence {E/'^OM)} Denote the space of differential k-forms on M by ^^(M). Let {J*(M), d} be the de Rham complex of M with complex coefficient and HT(M) the de Rham cohomology groups. If we denote A^(M) = A^(CT(M)*), then^^(M) = r(A\M)). Let FP(Ak(M)) be a subbundle of A^(M) defined by FP(A_ = { PROPOSITION 2.3.1. The collection {F^^^} is a decreasing filtration of the de Rham complex. PROOF. We only need to show that d(FPjk) �p_^k+i For any cMXi，...，Xp，〒i，...,〒k_p+2) � � � ,...，〒 ='gViri = 1 iMXi,.",Xp, i’..., i k_p+2) A A + E (-i)i〜（[〒i,�j]，Xi，...，Xp，〒i，〜〒i，...，〒..， 〒） l^i • • • • • • •參 會• it 4 ]f i F^^P D FiJP D ... � FPjP �P+i_^ =P {0} where the arrows represent the differential maps d:P_>^k — pjk+i， 0 < j < p + 1, 0 < k < p. . Since this differential respects the filtration, the de Rham cohomology groups H*(^ *) inherits a bounded decreasing filtration FPlT(j*) = image of inclusion H*(FP^*)^H*(^*). Define the associated graded module Eo'^ by EoP，q = pjp+q/FP+i_>^p+q. Then a sequence {Er^'^} associated with the filtration in can be constructed by the following proposition. THEOREM 2.3.2. For the filtered complex {¥^^ P+q，d}, there exists a sequence of differential bigraded modules {E^'^} (r > 0) together with differentials df ： E” — ErP + r,q-r + l and d^ = 0 such that .I _ 汽 ^ 叙 kcr d_ • Er “^ Ej> (1) HP+q(Er*'*) = Ef；!^ where IP+q(Er ' )=——：： ， � rnidr• j r：^ p-r,q-r+ Erl T">P,Q—E r (2) Ef’q = HP+q(FP_>T / FP+i_>r), and (3) E^ = FPRP+X^ *) / FP+iHP+q(_^ *). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 25 ‘ PROOF. We first introduce the following definitions ZrP,q = Fp_>^p+q 门 d-i(FP+r_^p+q), BfP'q = Fpjp+q 门 d(FP-rjP+q-i), Z=’q =门 ZrP’q = FP_>^P+q 门 ker d， BL,q = LlBrP'q = FP_>^P+q 门 im d. Since FPjP+Q is a bounded decreasing filtration, we have a decreasing sequence ZoP，q�zf’ Dq ... D z^,q�BS>,q�...�Bf,q�BoP，q. Moreover, d(Zr'''''"') = d(FP_r_^p+q_i 门 d_i(FP_>^P+q)) = B^'^ .....(*) Define E^'^ = Z” / 2^/''"' + B^'i^ and let r??'^ : Z^'' — E^'^ be the natural projection with ker)]� ,=q Z^/'^"^ + B^f. Using (*) and the property that d^ = 0，we have d(zr/,q-i + Bif) = d(zr/,q-i) + d(BP\^) 一 ^p+r,q-r+l —br_i r- r7p+r+l,q-r pp-r,q-r+l L ^T-l + ^r-l • Thus, the differential map d : Z” — Z^'Q-'+i induces a homomorphism dr ： E” — Er'q-r+l such that the following diagram commutes ZP，q J^. ^p+r,q-r+l I ”” 1 (.— jgP.q _^ gp+r,q-r+l Clearly, d^^ = 0. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 26 ‘ Consider the diagram Z/+l,q-l+BrP,q — Z/；!^ — ZfM ^ ZrP+r,q-r+l j ” I ”p/q 1 nr'^'' kerdr — E” 7 E"r,q-r+i i “ HP+q(E/’*) 4 0 where rj is the restriction of the map rf,'^ on Z|^;^, ir : kerdr — HP-^(E/'*，d� thise natural projection. We shall show that rj is surjective. Consider z E rj'^(ktx d^). Since d^ o rj = rj o d, d^. o rjz = 0 = rj o dz if and only if dz E ZrP_Ti'q-r + Bn'q-r+i c Z^T'''" + 6^1?^~"' = Z^''''" ^d(Z,'：,'), by (*). Equivalently, z € Z^_f^^ZPf. Thus, r/-^kerd^) = 2^：/'^"'+7^；?. Since Z^:J'^"' C keri??，^ we have ker d^ = 7y(Z';?). Next, we show that Zf''' + B^'^ = Zr。+’？门(rj”)'' (imd^). By the definition of d^ and using (*)，imd^ = r/P'\d(Zr'''''"')) = v^;H^f'^)- Hence, (77Pr'q)-i(imdr) = BrP,q + kerV;'q =B^ + zr/'q-i+BrPjiQ =BrP，q + ZrP_;l，q-l， since RPf C B”. By the definition of Z* ’ * and the decreasing property of F*J*, we have ^p+i.q-i 门 ^p.q _ ^p+i.q-i It follows that zp;?门（r7P/q)-1(imdr) = ZrP+\q 门（zr/'q-i +BrP,q) = z^'q-i+B^q. Now, we are ready to construct an isomorphism between HP'Q(E/，*，d� anE^^d；!^. Define a composite map 7 = w 0 rj ： Zl\^ — IP+q(Er*, *). As above, ker7 = Z,^;?门（r7^)-i(imdr) = zr'''"' +B^'^ Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 27 ‘ Since 7 is surjective, it induces the isomorphism yP.q HP+q(E/,*) = - = E^^ zri'q-i+B” To establish (2)，we define Z_Pi+i,q-i = p+i_^P+q and B_7 = d(FP+ijP+q'i). Thus, EoP,q = W’q / (Z_Pi+i,q-i + B_P!q) = FPjP+q / FP+iJP+q，as defined. Now, the differential d : F^^ P+q — FPj P+Q+i induces a homomorphism do ： Er - Er+1 and we have Ef'^ = HP+q {¥^^* / FP+i_^*). Define r)^J" : Zl^ ^ EL'' to be the projection with ktir]^^ = Z^''''' + B^'^ Also define the projection co : kerd — H*(j*). We have FPHP+q(J*) = im(HP+q(FP_^*) ^ W^\^*)) = co(pP^P^^ 门 kerd) = co(ZD. Since cj(ker ry。）= FP+ipP+X>^*)，w induces an isomorphism d„ : EL’q — FPHP+q(J*) / FP+iHP+X_^*). • In fact, kerd^ 二 rj';:'{o^-'{F^''R^'H^*))门 Z'^') = rj'^\Zi''门 dj* pi Z^)= r/p^q(z=+i'q-i+Br) = {o}. {E”} is convergent to E^'^ since the filtration Fm * (J*) is bounded. The proof is therefore complete. • DEFEVITION 2.3.3. The sequence of differential bigraded modules {Er^'^, dJ (r > 0) is called the spectral sequence associated to the filtered complex {pP_>^ P+q，d}. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 28 ‘ Let AP,q(M) = FP(AP+q(M))，JP,q(M) = r(AP,HM)) and let CP,qOVl)=AP,q(M)/AP+i,q-i(M), C^,q(M) = r(CP,q(M)). We also let 3j^ : 0"'%M) — C^'q+i(M) be the operator induced from the exterior derivative d : jP'q(M) — jP'q+i(M). 5j^ is called the tangential Cauchy-Riemann operator. Let HP，q be the cohomology groups ofthe complex {<^^'q(M), d^} and W-^ = Ef'^ Then jP'q(M) = r(FP(AP+q(M))) = pP^P^S (^^,q(M) = jP,q(M)/ jP+i'q-i(M) = E^P'q and ^M =�. Recall that t(M) = CT(M) / S is the holomorphic tangent bundle ofM with respect to the Cauchy-Riemann operator i^xM). Denote EP = /V(t(M))*. We shall show that CP，q(M) can be identified with C q(M, E P). First, we have the following straightforward result. PROPOSITION 2.3.4. EP is a holomorphic vector bundle with respect to the differential operator \, : r(EP) — r(EP0S') defined by i^ p _ 二 _ X) (-l)V(YUj,Ui, ...,Uj,…，...,u ) i = l for^Er(EP), Ui,…，Up G r(t(M)), YES, Ycp = (^,^)(Y) and Yu, = (‘)Ui)(�). Then, we can define a differential operator 恭:C\M, E^) — C^^\M, E^) as in Definition2.2.1. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 29 ‘ PROPOSITION 2.3.5. CP,q(M) s CHM,EP) and \ip = {-lf^l,ip for^ G ^^,q(M). PROOF. Define a map y^ : AP,q(M) — CHM,EP) by (7P<^)(u,,...,Up,Y1,...,Yq) =(yv)(^Xi)，...’^Xp)，〒i，...，〒q) =-(Xi，...，Xp，〒i，...，〒q) for ail It follows that Cq(M，EP) = A"(M) = cp，q(M). Ap+i,q-i(M) To prove the equality, the diagram AP,q(M) ^ cq(M,Ep) = Cp'q(M) 1 d I ^EP j ^M AP,q+i(M) 二' cq+i(M，Ep) = cp'q+i(M) should obey d^y^i/^ = (-iyy^d^/^ = (-Ifd^y^xP, for • E AP,q(M)^ Since d^ is induced by the operator d, y^ 0 d = d^ 0 7P. For any v^€AP’q(M)o，X”."，Xp G CT。（M) and Yp.",Yq+i G S^, Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 扣 ‘ (^fV)(WXi)”..’WXp)，〒i，...，〒q+i) = E(-l)i + i�i(7P^(Xi)，〜。(Xp)”"，〒i，...ii，...，〒。 i+” + l‘i£q“（]”(7P^^i)，...，^P),[�i，〒j]，〒”...ii，..4，...，〒 +q i) = E(-l)i + l�iWXl，...，Xp，〒i，...’.i，."，〒。 i)+) + l�SpJ-”"^l，...，Xp，[〒i，S，〒i，...ii，...，V..，〒。 l)+ = (-lW(Xi’...，Xp’�i’...，〒 +q i) =(-l)P7P"W(Xi)”..，。(Xp)，〒i，."，〒q + i). Hence, d^ Denote ^ = Ef，。，H�=£2'’。， < 1k < p. We know from Theorem 2.3.2. that ^ are the cohomology groups of the complex {义’ dJ, where d, : E” — Ef+i，。is induced by the operator d ： Z” — zf+i’�. Observe that Y = H^ ^ is the cohomology groups ofthe complex {C"''(M), dJ and �(M =) r(Ck,o(M)) ^ r(CO(M, Ek)) is the holomoiphic cross section of E^ DEFEVmON 2.3.6. The complex {兄 dJ is called the holomorphic de Rham complex and Ho^ is the holomorphic . de Rham cohomology groups. By Proposition 2.3.1., d>q(M) C >q+i(M). Thus {_>^,q(M), d} is a complex. The cohomology groups of this complex is denoted by H^'^ Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 31 ‘ PROPOSITION 2.3.7. The sequences of cohomology groups 0 — Ho' — H'/ii — Hk-i,i — Ht,i —... are exact. PROOF. As in Proposition 2.3.5., we have the short exact sequences 0 — ^'^M) — f,q+i(M) — c^-i'q+i(M) — 0. It induces the exact sequences of cohomology groups H'/ — Hk;i,i — Hk-ii — Hk;i —... where H^^ = { ^-i,o(M) = d^'^ • EXAMPLE 2.3.8.网 Suppose M is a complex manifold. Then HP,Q are clearly the Dolbeault cohomology groups and Ho^ are the usual holomorphic de Rham cohomology groups. 2.4. The Levi form Suppose that M is a CR manifold with CR structure S and real expression (H(M), J), where H(M) is a subbundle of T(M). Define a quotient subbundle G(M) = T(M)/H(M) of T(M) and a natural projection ir : T(M) — G(M). For any X，Y G r(H(M)), we define co : r(H(M)) x r(H(M)) — G(M) by co(X,Y) = 7T([X,Y]). 0) gives a smooth cross section of G(M) ® A^(H(M))* in view of the following proposition. PROPOSITION 2.4.1. For any X，Y E r(H(M)), (1) co(X,Y) = -w(Y,X), and (2) o)(fX,Y) = fco(X,Y) where f is any smooth real-valued function defined on M. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 32 ‘ PROOF. (1) is obvious since 7r([Y,X]) = -7r([X,Y]). For (2)，o;(fX,Y) = x([fX,Y]) =7T(f[X,Y] -fYX) =fco(X,Y) • PROPOSITION 2.4.2. For any X，YEr(H(M)), co(JX,JY) = w(X,Y). PROOF. We first show that [JX,JY]-[X,Y] G r(H(M)). For any X, YGr(H(M)), X-iJX, Y-iJY E F(S). Since M is a CR manifold, we have [X-iJX,Y-iJY] = [X,Y]-[JX,JY]-i([X,JX]+[JX,Y]) E T(S). Thus, [JX,JY] -[X,Y] G r(H(M)). Itfollows that 7r([X, Y] - [JX,JY]) =0. We have o)(JX,JY) = co(X,Y). • DEFEVITION 2.4.3. The Leviform at a point p G M is the map ^^ : Hp(M) x Hp(M) — Gp(M) defined by ^(X,Y) = cOp(JX,Y) for X, Y G Hp(M). Obviously, this definition is independent of the sections X，Y G r(H(M)). PROPOSITION 2.4.4. At each p G M, the Levi form SE^ satisfies the following properties: (1) iSp(JX,JY) = igp(X,Y), and (2) igp(X,Y) = ^(Y,X) for any X，Y G Hp(M). PROOF. The proof directly follows from Proposition 2.4.2. Indeed, i^p(JX,JY) = co(JJX,JY) = co(JX,Y) = igp(X,Y). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 33 ‘ Moreover it is symmetric since igp(X,Y) = w(JX,Y) = -co(Y,JX) = -w(JY,JJX) = o)(JY,X) = ^p(Y,X). • We extend 5E^ to CHp(M) by complex linearity. Then we have the following proposition. PROPOSITION 2.4.5. For each p E M, (1) ^p(u,v) = ^(v,u), (2) ^(n,v) = ^p(u,v), where u,vG CHp(M). PROOF. Let u = X-iJY, v = Z-iJW, X，Y, Z, W G Hp(M). Since ^^ is complex linear, the results are obtained easily from Proposition 2.4.4. • In the following, we shall examine the Levi form for an embedded CR manifold. As defined in Definition 1.2.7., the quotient space CTp(M) / CHp(M) can be identified with the orthogonal complement of CHp(M) = H^(M) 0 Hp'^(M) with respect to the Euclidean metric on CT(M). In this case, CT^(M)/CH^(M) is identified with the complexified totally real part CGp(M) of the tangent bundle. DEFEVITION 2.4.6. The Levi form of an embedded CR manifold M at a point p E M is the map 5Ep : Hp'^(M) — Gp(M) defmed by ^(Up) = ^7Tp([u,S]p), uGHi,�(M). Here, T^ : CTp(M) — Gp(M) is a natural projection. By Proposition 2.4.5., ^p(Up) = <^p(Up). Hence, its image is in Gp(M). Moreover, i^p(Up) = i^p(Up) for u E H^-^(M). Denote the orthogonal complement of Tp(M) in Tp(C") by Np(M). It is called the normal space of M at p. We define the Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 34 ‘ M is given by the defining function {p^, ..., p^} where d is the real codimension. Let [Vp^(p),...，Vp^(p)} be an orthonormal basis for Np(M). Then 开卩 is given by ^p(v) = Y： (dP.(P))(v)-Vp,(p) f = i for V E Tp(e). DEFEVITION 2.4.7. The extrinsic Leviform of M at p is the map iE^ : Hp'^(M) — Np(M) defined by ^ 二 ^p o J ° 4. Indeed, since Hp'^(M) and Hp^'^(M) are J-invariant, ^(Up) = lfp{J[u,i]p} foruGHi'o(M). ^ .� THEOREM 2.4.8. Suppose that M = { f E C" : p^(f) = ... = p^(f)} is a smooth embedded CR manifold with real codimension d, 1 < d < n. Let p G M and {Vp^(p), ..., Vp^(p)} be an r^ orthonormal basis for Np(M). Then the extrinsic Levi form ^^ is given by ^(Wp) = E t ^w,wJvp,(p) “1 j.k=i af.3f^ for Wp = t w,^ G H^'^M). k = 1 0 fk Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 35 ‘ PROOF. As above, ^(Wp)=l^p{J[W,W]^} =^E(dp,(p))(J[W,WyVp/p). Then the £th component of ^(Wp) = • (dp,(p))(J[W, W]) L 1 = -a*dp,(p))([w,wy, where J* : Tp* (C") — Tp* (C") is the dual map of J. Since J “ d = ia-i5，theahcomponentof«^(Wp)� =1((5 -5)p^(p))([W, W]. 2 p Recall that the Cartan formula for the exterior derivative of the l-forai - (d - d) p^ gives 2 l(do-a)p,)(w,w) = lw((a-a)p^)w -w(o-a)p,)w -i((a-a)p,)([w,w]). � •• z SinceWp G Hp'°(M), We have developed a formula for the Levi form in terms of the complex hessian of a set of defining functions for M. Now, we ftirther suppose that M is a generic embedded CR manifold with dim^M = 2 n - d，1 < d < n • By Lemma 1.3.1., we can choose suitable coordinates (z = x + iy, w) G C^ x C""^ such that p E M is the origin and the defining Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 36 ‘ functions for M are p^(z, w) = y^ -h^(x, w), 1 < i < d, with h^(0) = 0 and Dh^(0) = 0. Clearly, Vp^(0) = — andhence {Vp^(0), ...,Vp/0)} isanorthonormal ^ye basis for the normal space No(M). Hence No(M) can be identified with M^ by the map y = (yi，...，yd) - E y./- ^ N�(M). “1 oy^ Furthermore, we can identify Ho'^(M) with C""^ by the map w = (Wi,...,w„_,) - 2 w,^ G Ho^''(M). k = i (^w^ COROLLARY 2.4.9. Suppose M = {(z = X + iy, w) G C^ x C""^ : y = h(x,w)} is a smooth generic embedded CR manifold where h : R^ X C""^ — M^ is smooth with h(0) = 0 and Dh(0) = 0. The extrinsic Levi form at the origin is given by ^(W) = (yi”"，yd)eRd n-d fh(0) _ where y^ = J^ _ll^w^w^, 1 < i < d, for W = (w^, ...,w^_^) G C"-^ j,k = i dw.dw^ EXAMPLE 2.4.10. M = {(Zi，Z2，Wi，W2) G c;4: y^ = q^(w, w),y^ = q!(w, w),w E C^} is a quadric submanifold of codimension 2 in C^ where z^ = Xj + iy^, z^ =x^+iy: and q = (q^, q2) : C2 X C^ — C2 is a quadratic form. From Corollary 2.4.9.，the Levi form of M at the origin is given by w 二（w”W2) 4 J2 8 q(2 w w^. j,k = i d w. dw^ It is in fact the quadratic form q(w,w) of M restricted to {(w, w): w E C^}. From Theorem 1.3.8., we know that M can be expressed in different forms depending on the quadratic forms q! and ^. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 37 ‘ If q^ and q^ are linearly dependent over R, then M = {y^ = q!(w, w)，y^ =0}，and hence the image of the Levi form of M at the origin is a subset of a straight line in R^. If q^ and 七 are linearly independent over R, M can have 3 different forms respectively. The fu:st form is M = {y^ = | w^ p, y^ = | w^� an} d the image of the Levi form is the closed quadrant { y^ > 0’ y^ > 0}. When M = {y^ = | w! p, y^ = Re (Wi^}，the image of the Levi form is the open half plane {y^ > 0} together with the origin. Lastly, M = {y^ 二 Re(Wi&2)，5^2 : Im (w^Wj)} and the image of the Levi form is the whole plane R2. In all these cases, the image of the Levi form is a convex cone in N。（M). 2.5. Strongly pseudo-convex manifolds Let M be a manifold and H(M) a subbundle of T(M). As before, we have defined a quotient bundle G(M) = T(M)/H(M) and a skew-symmetric bilinear map co. DEFEVITION 2.5.1. The subbundle H(M) of T(M) is called a contact structure if (a) dim^G = 1 and (b) cjp is non-degenerate for each p E M, i.e., for every X E Hp(M), cj(X, Y) = 0 for all Y G Hp(M) implies that X = 0. The manifold M together with the contact structure H(M) is called a contact manifold. It can be shown that a contact manifold is of odd dimension [Bl]. Suppose dim^jM = 2n_l. We assume G(M) and hence G(M)* are trivial. Let 6 be a trivialization of G(M)*. It is a cross section of G(M)* such that 6^ 7^ 0 for all p G M. Thus, 6 is a 1-form on M and the subbundle H(M) is characterized by [K1] H(M) = {XE T(M) : 6(X) = 0}. Since d0(X,Y) = -0([X,Y]) for all X，Y G r(H(M)), we see that the restriction of d0p to Hp(M) is non-degenerate for all p G M. Hence the (2n - l)-form dA(ddy-^ is a volume form on M. Here 6 is called a contactform. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 38 ‘ DEFEVITION 2.5.2. A vector field X is called a infinitesimal contact transformation if [X，Y] G r(H(M)) for all Y G r(H(M)). It can be shown by Darboux Theorem [St] that there exists a unique infinitesimal contact transformation 专 such that e{i) = 1 and 专」d0 = 0. In fact, for X E r(H(M)), d6(^,X) =0 implies that ^([^X]) =0. Hence, [^,X]Gr(H(M)). Moreover, the assignment 6 >^ ^ gives a linear isomorphism from the set of all 4 trivializations 6 of G(M)* onto the set of all infinitesimal contact transformation ^ such that ‘ € Hp(M) for all p G M. Clearly, G(M) is spanned solely by ^ and T(M) = H(M) © G(M). PROPOSITION 2.5.3. Suppose H(M) is a contact structure arid G(M) is trivial. The Levi form at p E M is given by igp(X,Y) = -d0(JX,Y)-7T(fp), X,YGHp(M). PROOF. Since T(M) = H(M) 0 G(M), [JX,Y] E T(M) is decomposed into two components: [JX,Y] = [JX,Y]" + X^,for X,YE Hp(M), where [JX,Y]" istheH- component of [JX, Y] and X G R. We have X = 0([JX,Y]) and hence, 7r([JX,Y]) = 7T(0([JX,Y])-O = 0([JX,Y]).7T(?) = -d0(JX,Y).7r($). Therefore, ^p(X,Y) = -d0(JX,Y).7r(^p). • DEFEVITION 2.5.4. Let M be a CR manifold with CR structure S and real expression (H(M), J). Let G(M) = T(M)/H(M) be the quotient line subbundle of T(M). If the Levi form ^^ is definite at each p G M, i.e., ^p(X,X) = 0 for X E Hp(M) implies that X = 0, then S • is called a strongly pseudo-convex structure and M is called a strongly pseudo-convex manifold. t Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 39 ‘ Clearly, if M is a strongly pseudo-convex manifold, then M is a contact manifold. PROPOSITION 2.5.5. Suppose M is a strongly pseudo-convex manifold. Then G(M) is trivial and there is a trivialization 6 of G(M)* such that at each p E M, the Heraiitian form -d0(JX, X) is positive defmite for X E Hp(M). PROOF. For any p E M and any non-zero X E Hp(M), since X^ is definite at p, iSp(X,X) provides a distinguished ray of the line Gp(M). Hence, G(M) is trivial. Choose a trivialization 6 of G(M)* which is positive on the distinguished ray. Regard6' as a 1-form on M and take the corresponding field ^ on M. Then, Proposition 2.5.3.‘ implies that -d0(JX, X) is positive definite. • DEFmTION 2.5.6. The trivialization 6 of G(M)* defined above is called the basicform. The corresponding infinitesimal contact transformation ^ satisfying ^^ ¢. Hp(M) for each p E M is called the basic field. The following result is obvious. PROPOSITION 2.5.7. Let d be a basic form. A 1-form r] is a basic form if and only if rj = f6, for a positive function f. Note that every strongly pseudo-convex manifold is orientable because^A(d^)""^ is a nowhere vanishing (2n - l)-form. • Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 40 2.6. Strongly pseudo-convex real hypersurfaces We tum to study the strongly pseudo-convex real hypersurfaces in C". Recall that a real hypersurface of C^ is a generic embedded CR manifold. Let M, be an n-dimensional complex manifold with complex structure S'. Let f be a real-valued function defined on an open set U of M^ such that dfp j^ 0 for all p G U. DEFESmON 2.6.1. For each p E U, we define a subspace S(f)p of Sp by S(f)p = {XGS； : df(X) =0}. We also define a Levi form <^(f)p on S(f)p by — ig(f)p(X,Y) = (aaf)(X,Y), X,YGS(f)p. Let f-i(0) 5^ 0 and M = f"^0) be a real hypersurface. Let S be the CR structure of M and (H(M),J) its real expression. Clearly, S(f)pA^^ = {0} and df([X,Y]) = X(df)(Y)-Y(df)(X)-(d2f)(X,Y) = 0 foranyX,YES(f)p. Hence S(f)p = Sp，pEM. We define a 1-form 6 on M by 6 = h*df = -ii*df where i : M ^ M, is an injection. Then 6 is a real l-form and it is a trivialization of G(M)*. Now, the Levi form ^(f)p can be expressed in terms of d. PROPOSITION 2.6.2. ForanyX,YGS(f)p, i^(f)p(X,Y) = -id0(X,Y). PROOF. The result is by the fact that d = d +d, d^ = 0 and d^ = 0. • By this proposition, we can define the strongly pseudo-convex real hypersurface. DEFEVITION 2.6.3. A real hypersurface M in C" is called strongly pseudo-convex if a real l-form B is chosen so that the Levi form is definite at each point in M. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 41 ‘ If p is a point on a real hypersurface M = {zG C" : f(z) = 0} with V f(p) I = 1，then by Theorem 2.4.8., the extrinsic Levi form is given by »• • ^(W)= EE^^w.w, Vf(p) j = i k = i dfj3fk forW^ = ^w,^GH;'^M). k = l ^Sk The condition that | V f(p) | = 1 can be obtained by multiplying f by a suitable scalar. In this case, the normal space Np(M) is identified with R by the map t^tVf(p), t E R. Thus, the extrinsic Levi form is identified with the restriction of the complex hessian of f to Hpi'o(M). By Definition 2.6.1., we have tiie following result. PROPOSITION 2.6.4. A real hypersurface M of C" is strongly pseudo-convex if there exists a defining function f for M such that at each p G M， E -^^w. w, > 0 for all W^ = ^ w,A G W,�(M). j,k = i 5fj3fk 卜1 a“ THEOREM 2.6.5. Suppose M is a real hypersurface of C" that is strongly pseudo-convex at a point p E M. Then there is a biholomorphic map (;i3{Mf|U} is a strongly convex hypersurface in PROOF. We choose the coordinates (z,w) E C x C"] so that p is the origin and by Theorem 1.3.3., M = {(z = x + iy,w) G C x C"'^ : ”h(x，w)}, where h : R x C""^ — R is smooth, h(0) = 0，Dh(0) = 0 and there are no second-order pure terms in the expansion of h about the origin, i.e., a2_ 二 a^h(0) 二 0. 5wj6w^ dx. dw^ ‘ Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 42 ‘ Let f(z,w) = y -h(z, w). Then we have f(z,w) =y-x! i^Wj< + 0(3) =y + "f i^w.w,.0(3), j.k = i dw.dw^ j.k = i aWj5Wj^ where 0(3) denotes the terms that vanish to third order in x and w. By hypothesis, J^ ^ ^^^^ WjW^ > 0. j,k = l dwj5w^ Let f = f + 2f^. Here f is also a defining function for M. Then f(z,w) = y+2y^+ ^ 竹�一)WjW + ,0(3) j.k = i dw.dw^ = y-Re(z2)+|z|2+nf ^f(0_) WjW,+0(3). j,k = i dWjdWj^ Define the map cp : (z，w) G C x C""^ ^ (z, w) by A . 7 Z = Z - lZ^ w = w Clearly, (p is a local biholomorphism aild it preserves the set { (0，w) : w G C""^ }. If z = X + iy, then y = y -Re(z^) and | z |^ = | z |^ + 0(3). Let M = EXAMPLE 2.6.6. Let M be a pseudo-convex domain in C^ with a smooth boundary d M. Suppose the surface d M with coordinates (z, w) is defined by a smooth real-valued function f = Rew + z 8 + i£ z 2Rez6 7 such that df 9^ 0 on M and 'f = 0，on aM, « f < 0, inside M， f > 0, outside M . For each p E 5M, we define Hp'^(3 M) the subspace of the tangent space at p consisting of the tangent vector X G Hp'^(5 M) given by X=a!+b_A_ with < 5f, X> = 0. dz aw p Clearly, — = -. Thus, b = -2a ^ and X = a A-2a ^ 各. aw 2 dz p dz dz p ^w Lw 一 Moreover, ^'� =_|lL. = _J^ = 0 and _^X = 16 | z |^ + 15Rez^ Hence, d z d w dzdw d w d w dzdz the extrinsic Levi form at p is ^(X) = -[(16 | z |^ + 15Rez^) | a |']Vf(p) which is either positive or negative except at z = 0. Thus, the surface d M is strongly pseudo- convex at every point except the line { (z，w) E C^ : z = 0 and Re w = 0 }. Therefore M = {(z, w) E C2 ： f(z) < 0 } is a domain of holomorphy. Let U be a neighbourhood of the origin and cp a holomorphic function defined on U and vanish at the origin. It can be shown that there exist two points (z^’ w^) and (z^,^v^) in U such that cp(z^,w^) = 0 = (p(z^,w^) and f(z^,Wj) > 0, f(z^, w^) < 0. In this case, it is impossible to introduce holomorphic coordinates relative to which d M is convex in U. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 44 2.7. Canonical affine connections Let M be a strongly pseudo-convex manifold with structure S and M is of real dimension 2n - 1. d and ？ are basic form and basic field on M respectively. (H(M), J) is a real expression of M and G(M) is the quotient line subbundle of T(M) spanned by f. Let 0) = - d6. Then f� o=; 0. We extend the complex linear map J to a complex linear structure map of T(M) to T(M) so that J ^ = 0. PROPOSITION 2.7.1. At each p E M， (1) J2X : -X + 0(x)r (2) o;(JX,JY) = co(X,Y), forX，YGTp(M). PROOF. Since T(M) = H(M) © G(M), any X E T(M) can be decomposed as X = Xjj + X^, where X^ is the H-component of X and X E M. Then JX = JXy + XJ^ = JX^ andthus J^X = PX^ = -X^. Moreover,入=6(X). Wehave J2X = -X + 0(X)^, for any X E Tp(M). Toestablish(2)，forX，YGTp(M), co(JX,JY) = c^(JX^JX^) = o:(X^,Y^), where Xj^, Y^ are the H-components of X and Y respectively. On the other hand, since 专」CO = 0，co(X,Y) = co(XH,YH). Therefore, cu(JX,JY) = co(X,Y). • Define g(X,Y) = co(JX,Y), X,YG Tp(M). Then g is symmetric and g(JX,JY) = g(X,Y). For each p G M, the restriction of gp to Hp(M) is clearly positive definite. Let X,YG r(CH(M)). Since, CT(M)= CH(M) © CG(M), [X,Y] can be decomposed into the CH- and CG-components, denoted by [X, Y]^ and[X, Y]^ respectively. Then we have [X,Y]^ = co(X,Y)^ Since CH(M) = S © S, [X,Y]^ is further decomposed into two components: [X’ Y]^j = [X，Y]^ + [X，Y]g. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 45 ‘ THEOREM 2.7.2. Suppose M is a strongly pseudo-convex manifold with structure S. 6 and 专 are basic form and basic field on M respectively. (H(M), J) is a real expression of M. There is a unique affine connection V : r(T(M)) — r(T(M) 0 T(M)*) on M satisfying the following conditions: (a) The contact structure H(M) is parallel, i.e., Vxr(H(M)) C r(H(M)) for any X E r(T(M)). (b) The tensor field 态，J and co are parallel, i.e., •态=VJ = Vco = 0. (c) The torsion T of V satisfies the equalities: (1) T(X,Y) = -a;(X,Y)^ and (2) T(^,JY) = -JT(^Y) for X,YEHp(M), pGM. PROOF. (Uniqueness) Let V be the required affine connection. We extendV to a differential operator of r(CT(M)) to r(CT(M) ® (CT(M))*). Since H(M) is parallel and VJ =0， V^(T(S)) C r(S) and V^(T(S)) C r(S) for any X G r(CT(M)). We first show that V satisfies the following conditions: For any X,Y,ZG F(S), (i) V3^Y = [X,Y]3. (ii) co(V^Y,Z) = Xc0(Y,Z)-c0(Y,[X,Z]3). (iii) •专 Y = L^ Y + T^ Y，where L^ is the Lie derivation and T^ = - - J • L^ J • Z^ To establish (i), we have T(X,Y) = V^Y - VyX - [X,Y]. On the other hand, T(X,Y) = -o;(X,Y)? = -[X,Y]^ = [X,Y]^ - [X,Y],by (c). Hence, V^Y - V^X = [X,Y]^ = [X,Y]3 + [X,Y]3. Therefore, V^Y = [X,Y]^ and •yX = [X,Y]3. For (ii), since Vco = 0 by W, Xo;(Y,Z) =co(VxY，Z)+co(Y，VxZ). Thus，using®， we have the result. For(iii),sinceV? = 0，V^Y = T(^Y) + [?,Y] andV^(JY) = T({,JY) + [^,JY]. Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 46 ‘ By 00 and (c), VJ = 0 and T(?,JY) = -JT(?，Y). We have (L^J)Y = L^(JY) - JL^Y :[$,JY] - J[^Y] =V^(JY) + JT(f,Y) 一 JV^Y + JT(^Y) =2JT(^,Y) It follows that T(J,Y) = -ij((LJ)Y) = T^Y. Hence,V^Y =T(^Y) + [^Y]= J^ T^Y + L^Y. --. Since M is a contac. t manifold, co is non-degenerate, i.e., for all Y G S^,o)(X, Y) = 0 implies that X = 0. Moreover, V? 二 0 and V^W = •乏两 for all Z,WE r(CT(M)). V is therefore uniquely determined. (Existence) It suffices to show the following: (iv) JT^ = -T^J. � (v) If Y E r(S)，then L^Y + T^Y G r(S). (vi) L^ CO = 0. (vii) o)(T^X,Y) + co(X,T^Y) = 0 for X,YE Tp(M). (viii) For X,Y,ZG T(S), Xc0(Y,Z)+Ya)(Z,X)+c0(Z,[X,Y])+c0(X,[Y,Z]3)+c0(Y,[Z,X]s) 二 0. For (iv), note that for any X E Tp(M)，since f� d=0 0， (L^^)(X) = L^(eX)-6L^X =^e{X)-6{[^,X]) =^s{x)^dd{^,x)-^d{x) =0. JT^ + T^J = 0 if and only if J(LJ)+(LJ)J = 0. In fact, for any X E Tp(M), J2X = -X + 0(X)r Hence, Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 47 ‘ • [J(LJ) + (V)](X) = J((LJ)(X)) + (V)(JX) =J(L^(JX)-JL^X) + L^(J^X)-J(L^(JX)) =L^(-X+6>(X)0 - J'L^X =-L^X + (L^(0(X)))^ + L^X - 0(L^X)f =(L^WX))U - (L^WX))U + ((L^0)(X))? =0. Making use of (iv), we have J(L^Y + T^Y) = L^(JY) - (V)Y - T^JY =L^(JY) - 2JT^Y - T^JY =L^(JY) + 2T^JY - T^JY =L-“JY) + TJY =i(L^Y + TJ). Hence, L^Y + T^Y C r(S), proving (v). (vi) is easily obtained since co = -dO, L^ 6 = 0 and L^d = dL^. For (vii), we use (vi) and the fact that co(JX, JY) = co(X,Y). Hence,L^o;(JX,JY) =0 implies that co(L^JX,JY)+o)(JX,L^JY) = co(JL^JX, -Y)+a;(-X,JL^JY) =ico(T^X,Y) + lo;(X,T^Y) =0. (viii) is clear since dcj = -d^6 = 0. Define a bilinear map V : r(CT(M)) x r(CT(M)) — r(CT(M)) such that 1. V satisfies (i), (ii) and (iii). 2. For any ZEr(CT(M)), V^? =0. 3. For any Z，W E r(CT(M)), V^W = V_W. Then V is the complexification of an affine connection. It is left to check the conditions (a), (b) and (c). Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 48 ‘ Now, VxO) = 0, T(f,X) = V^X - V^^ -L^X = T^X, and T(X,Y) = -o)(X,Y) for X,YESp. By (V)，V^r(S) C r(S). Moreover, •一 = T^co = 0. By (ii), we have c0(X,[Y,Z]3) = Yo)(X,Z) - co(VYX,Z) co(Y, [Z,X]s) = -Xco(Y, Z) + o)(V^Y, Z) Hence, by (viii), o)(T(X,Y),Z) = 0. Since w is non-degenerate, T(X,Y) = 0. Therefore, V is the required affine connection. • DEFEVITION 2.7.3. The affine connection defined in Theorem 2.7.2. is called the canonical affine connection of the pair (M, ^). For any X, Y, Z G Tp(M), the curvature R satisfies the equality R(X,Y)Z = V^VyZ - V^V^Z - V[x,Y]Z. Clearly, for any Z G Tp(M), R(X, Y)Z C Hp(M). PROPOSITION 2.7.4. For any X,Y,Z,WG Hp(M), (1) R(X, Y)JW = JR(X, Y)W. (2) g(R(X, Y)Z, W) + g(Z, R(X, Y)W) = 0. (3) R(X，Y)Z+R(Y,Z)X+R(Z,X)Y =-o)(X,Y)T^Z-co(Y,Z)T^X-co(Z,X)T^Y. PROOF. (1) is easily obtained since VJ = 0. For (2), since g is bilinear and symmetric, it is equivalent to show that g(R(X,Y)(Z+W),Z+W) =0. We shall show that g(R(X, Y)Z,Z) =0. The condition (ii) in Theorem 2.7.2. gives the equality co(V^Y,Z) = Xa;(Y,Z)-a:(Y,[X,Z]) for X,Y,ZE Hp(M). Hence, by using the Jacobi's identity, we have Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 49 ‘ g(R(X,Y)Z,Z) =cj(JR(X,Y)Z,Z) =�(R(X，Y)JZ，Z) =w(VxVYJZ-VYVxJZ-V^x.Y]JZ, Z) =Xco(VyJZ,Z) - co(VyJZ, [X，Z]) -Yo;(VxJZ,Z) + co(VxJZ, [Y, Z]) -[X,Y]co(JZ,Z) + co(JZ,[[X,Y],Z]) =XYo;(JZ,Z) - Xco(JZ, [Y, Z]) - Y(JZ, [X，Z]) + o^(JZ, [Y, [X，Z]]) -YXco(JZ, Z) 4- Yco(JZ, [X, Z]) + Xco(JZ, [Y,Z]) - co(JZ, [X, [Y,Z]]) -[X,Y]o;(JZ,Z) + c^(JZ,[[X,Y],Z) =0. For (3), the computation is straight-forward by using the torsion condition in Theorem 2.7.2. and dco = 0 • COROLLARY 2.7.5. Let X,Y,Z,WG Sp. Then (1) R(X,Y)Sp C Sp. (2) R(X,Y)Z = R(Z,Y)X. (3) g(R(X,Y)Z,W) = g(R(Z,W)X,Y). PROOF. Since JR(X,Y)V = R(X,Y)JV = iR(X,Y)V, (1) is proved. For (2)，we observe the equality (3) of Proposition 2.7.4. that R(X, Y)Z, R(Y,Z)XESp, R(Z,X)YGSp. But for any VES^, T^V = T(^V)GSp bythe fact that -JT^V = T^JV = T(^,JV) = iT(^Y). Hence, R(X,Y)Z4-R(Y,Z) = 0. Therefore, R(X,Y)Z = R(Z,Y)X. To establish (3), we have g(R(X, Y)Z,W) = g(R(Z,Y)X,W) : -g(X,R(Z,Y)W) =g(X,R(Y,Z)W) = g(X,R(W,Z)Y) = g(R(Z,W)X,Y). • Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 50 ‘ DEFEVITION 2.7.6. Let {e^, ...,e^_i) be any orthonormal basis of Sp with respect to the metric g，i.e., gCepij) = b... Define the Ricci operator R, : CH(M) — CH(M) by R*X = -ixi R(ej,e.)JX j = i for XE CHp(M). Clearly, the operator is well-defined, i.e. R(ej,e.) is independent of choice of the orthonormal basis. The Ricci operator has the following properties. PROPOSITION 2.7.7. (1) R* is a real operator, i.e., R* X = R* X for X G Sp. (2) R, S C S. (3) g(R,X,Y) = g(X,:^). PROOF. It suffices to show that R(Cj,^) = R(ej,ep. The equality holds since • Y = VlY, V_Y = VTY and [x,Y] = [X,Y]，for X,Y E r(S). Then we have 入 X X 入 S7X = iE R(epS:)JX = iE R(F,e3)JX = -iE R(e^,i;)JX = R*X. j=i J j=i j=i (1) is thus proved. It is easy to check that JR* X = iR* X for any X E S^, proving (2). n - 1 n - 1 For (3), we have g(R,X,Y) = g(-iE R(ej,^)JX,Y) = g(E R(Cj,ipx,Y) j=i j=i =_g(x/£ R(ej,Z)Y) = g(X, -1¾ R(ej,i;)JY) = g(X,R,Y). • j = 1 j = i Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 51 2.8 Green's theorem Recall that 6A(d6y~^ is a volume element on M. Denote it by dv. Letp G M and {e^, ...,e^_j} be any orthonormal basis for Sp. For any a E F(S*), we define a function 5"a on M by (� =p -E (v)(# j = l J We also define a function 5'P by (^)p = "E (V)(ep， for/3Er(S*). j = i J Define p by ]S(X) = M^ for all X E S^. PROPOSITION 2.8.1. S'0 = ¥^ on M. PROOF. ForanypG M, (5"幻卩=^.-¾ (VJ)(^) = -¾ [ej0(^))-^(V,^)]. j = 1 ‘ j = 1 ‘ Hence, ^¾ = -¾ [Hj(/3(e.))-/5(V_e.)]=柳)?. • j = i J LEMMA 2.8.2.岡 Let fG CF(M) and a G T(S*). (1) ff*dv=d(f*^�dv). n-l (2) 5"5:.dv = -da, where a is the 2(n — l)-form defined by a = J^ a(H^eJ dv. j = i J J PROOF. Since L^d = 0, we have L^dv = 0 and it follows that L^(fdv) = Jf.dv. On the other hand, L^(fdv) = d(?�（fdv))+?�d(fdv = d(ff) �dv),proving(l). Define A^ a(l, l)-tensor field by A^ = L^ - V^ for X G r(CT(M)). Since Vco = 0， V 6 = 0 and V dv = 0. Let { e^，...，e� ^ _} be an orthonormal basis for S^. -• Chapter 2: Differential Geometry of Strongly Pseudo-convex Manifolds 52 ‘ Then we have (Lxdv)(ei，."，en_i) n-l =X(dv(ei, ...,e^_i)) - Y^ dv(� ...,h^Q.^,...,t^_^)” j = i n - 1 n - 1 =X(dv(e^,...,e^_i)) - ^) &v{t^,...,V^t., ...,e^_^) - ^) dw(t^,--.,A^&., ...,e^_^) j=i j=i n-l =(Vxdv)(ej,...,e^_i) - J^ dw(Q^,...,A^Q., ...,e^_j) j = i n-l =Y, dv(e^, ...,Ax6j, ...,e^_i) • j = i = -(traceAx)dv(ep...,en_i), for all X G r(CT(M)). We pick X E T(S) such that a(Y) = g(X,Y) for all Y E S^. Then T(X, Z) G CGp(M) if Z G CHp(M). Moreover, T(X,态）G CHp(M). Hence, 二n-l 一 一 traceAx j = i X) [g(Axej,$)+g(Axij,ej)+g(Ax^,^)] =E [g(-V^X-T(X,e.),e.)+g(-V-X-T(X,ep,e.) j=i J J • + g(-V^X-T(X,f),0] • ，；） =-jE = l g(V ‘ =b" a Thus, b" o:-dv = (traceAx)dv = -L^dv = -d(X�dv =) -da. • Recall that if M is a strongly pseudo-convex manifold, it is orientable. We further • suppose that M is compact. Then we have the Green's Theorem as follows. THEOREM 2.8.3. (Green's Theorem) • ^f.dv 二 5" ordv = 5fdv = 0. •f M ^ M ^ M Chapter 2: Differential Geometry ofStrongly Pseudo-convex Manifolds 53 2.9. Canonical connections in holomorphic vector bundles Let E be a complex vector bundle over a manifold M. DEFEVITION 2.9.1. A differential operator D : r(E) — r(E®CT(M)*) is called a connection in E if D(fu) = fDu + u0df for all u G r(E)，fE CF(M). Denote the covariant derivative (Du)(X) of u in the direction of X G CTp(M) by D^u. The curvature of D is defined by K(X,Y)u = Dx(DyU) - DY(DxU) _ D[x,Y]u. Here K gives a cross section of E ® E* ® A^(CT(M)*). We fix an affine connection V on M. Then a connection D in E together with V defines the covariant derivative D : r(E;0(CT(M))J) — r(E;®(CT(M))^i). In particular, for LEMMA 2.9.2. (Ricci Formula) For any =(1)-⑵-⑶ + ⑷-(5) + (6) + (7) + (8) » On the other hand, K(X,Y)(^(Xj, ...,X,) : DxDyWXi, ...,X,)-DYDx<^(X,, ...，X�-D[x’Y]<^(Xi，...，X� =⑴-(9) - (14) _(DT(X,Y)P)(Xi，...，X� e = � ^(^1 ’ …， � -D^^YX ^(^1'…’ X j J=+^ l ^v^Y^j'…’ X f xP(Xi，...，X�-5 (DyDxaKXi，…，X� =(DY(D^(X，Xi，."，X� t =D^(D t 1 ，...， ，...， ...，X》） + j5 =2 l P(Xi VvxXjY , ...,Xp "Xj =) l Dx(v?(Xi VyXj, + 5^ ^(Xj, ..., V^Xj^, ..., VyXj, ..., X^) 1 + 1 t + 5^ P(Xi，..., V^VyXj, ..., X^) j = l �.. =(9)-⑶-(10) + (11) - (2) + (7) + (6) + (13). -;^WX”...’R(X,Y)Xj，...，X� j = i t t ="5^ <;^(Xi, ...,V^V^X., ...，x�+5 t + 5^ V^(Xi，...，V[x Y]Xj，...，X� j = l ， =(1) - (9) - (14) Therefore, the result is proved. • We further suppose that M is a strongly pseudo-convex manifold with basic field ^ on M. Chapter 2: Differential Geometry ofStrongly Pseudo-convex Manifolds 56 THEOREM 2.9.3. Let E be a holomorphic vector bundle over M and a Hermitian inner product <，> in E. Then there is a unique connection D in E satisfying the following conditions: (a) D^u = Xu = (&u)(X) for u G r(E) and X E r(S). ¢0 X n-l (c) Let { Cj，•..，e„_ ^} be any orthonormal basis for Sp. Then ^ K(e. ’ e.) = 0. j = i PROOF. (Uniqueness) Let D be any connection satisfying the conditions (a), (b), (c). The equation in (a) defmes D^. By (b), X n - 1 n - 1 E B(ej,epu + 5]Du((n-l)iO = 0 j=i j=i . n-l We have D u = (Du)(J) = _i_Y;B(e.,e.)u. ..…（*) � n -1 fr( � J ^ ) Therefore, D is uniquely determined. (Existence) Defme a bilinear map D^ : r(CH(M)) x r(CH(M)) ^ r(E) by D^(X,u) = D^u such that D^u = Xu and D^u is determined by X Therefore D satisfies condition (b) and the proof is complete. • DEFEVITION 2.9.4. The connection D defined in Theorem 2.9.3. is called the canonical connection of the holomorphic vector bundle E with respect to the Hermitian inner product <，> and to the basic field ^. PROPOSITION 2.9.5. Let X，Y E Sp and u,v G Ep. Then � K(X，Y) = K(X,Y) = 0. (2) =0. By (**)inTheorem2.9.3., for Z,WG CHp(M), -T(Z,W) Hence, hi this chapter, we introduce the Laplacian D^ in the harmonic theory. We shall prove its subellipticity and give the main theorem which is essentially due to Kohn. Then we show a duality theorem on the cohomology groups HP,q. Let M be a strongly pseudo-convex manifold of real dimension 2n - 1 and ^ is a basic field on M. Let E be a holomorphic vector bundle over M and <，> a Hermitian inner product in E. There is a unique canonical connection D of E with respect to the inner product and to ^. The covariant differentiation D is defined with respect to the canonical affine connection V. 3.1. The fundamental operators Recall that C^(M,E) = E ® A^S*. Since CT(M) = S 0 S 0 CG(M), the vectorbundle C^M, E) can be identified with a subbundle of E ® Aq(CT(M)*). Forany X G r(CT(M)) and PROPOSITION 3.1.1. For (^E^)(x,,...,x^.,) = E(-iy“c^,)(3^i，...X...,3^q+i). i = 1 ‘ PROOF. The result is clear by simply expanding D^ The inner product <，> together with the tensor field g induces an inner product in C%M, E) by the following way. Chapter 3: The Harmonic Theory 60 DEFmXION 3.1.2. Let Clearly, this inner product is well-defined, i.e., it is independent of the choice of basis for S^ • DEFEVITION 3.1.3. Define a differential operator ^ : C^^\M, E) — C%M, E) by (^^)(X^,...,X^) = -E(D,^^)(ipX,,...,X^) where cp E C^+^(M, E). It is not difficult to check that t^ = 0. PROPOSITION 3.1.4. For any ,XGSp. Then < \ PROOF. Making use of the equality (b) in Theorem 2.9.3.，we have < 5g^, \P > = “i�5,W < = •i<5^ = 1�,5^S{ n-l Since V-e. =Eg(Vi^，e�5 ‘ t = 1 'j n-l =E fi�(g(\ �, )) -g5 (\，\e,)e,} n-l =-Eg(\��5” “1 'j -.� we have -(3) - (4) = (6). � -5 i==2(V4(ei ‘ ) n-l =E {5<0^�� - •<> <^，Vei��•�} =(1) - (5). Therefore, we proved the equality. • DEFEVITION 3.1.5. The Laplacian D^ : ^(M, E) — P(M, E) is a differential operator defined by •E = ^E^ +¾¾. A solution (p G 0(M, E) of the equation D^ cp = 0 is called a harmonicform. The space of harmonic forms in C\M, E) is denoted by H'(M, E). Chapter 3: The Harmonic Theory ^^ Clearly, we have B^\ = \ D^ and Dg^ = ^^ D^. Using the Ricci operator R* : CH(M) — CH(M) in Definition 2.7.6.，we define an operator R, as follows: .. DEFEVITION 3.1.6. The Ricci operator R* : C^(M, E) — C^(M, E) is defined by (R,^)(X,,...,X^) =^v'(X^,...,R^X.,...,X) j = i for all PROPOSITION 3.1.7. The Ricci operator R^ is self-adjoint with respect to the inner product <，> in (^(M, E), i.e., • DEFEVITION 3.1.8. Using the curvature K of D, we define an operator K : P(M，E) — 0(M, E) by (KW(3^i”.jq) =EE(-ir^K(e,,Xp^(e^,X,,...i.,...,XJ. • i = l j = l J q .. PROPOSITION 3.1.9. The operator K is self-adjoint with respect to the inner product < , > in (^(M, E). PROOF. Using (2) of Proposition 2.9.5., we have, for any X, Y G S^, PROPOSITION 3.1.11. For any 9? G C^{M, E), n-l . (1) U^^ = -^ D^D-^ - qiD# + K^ + R^<^. j = 1 J J n-l � Og<；^ = "E D;jDe, + (n-q-l)iD# + K � °E^¥|DejV-AgDApQqf PROOF. First, we have (^^^)(X,,...,X^) : 二n-l _ 一 一 j= l"E(DA^)(e.,X,,...,X^ J ) n-l _ _ _ _ =-l{Dej((4W�3^1，...’3^q))-(4^Ve�3^”...，3^q) j=l J q 一 一 一 一 "E(^E^)(ej,x,,...,v^x,,...,x^)} ”..， � =-E{De,(j=l D W(3^1 3^q))-De.(:^(-l)J k=l k + l(hWk i^l”.. i ..,X)4 ) -(Dvjy(5^”...,3^q)-kg(-l)k + l(D�_ej€,3^1”..,^k”"，*q) A -g[CV)(*i，...，V,^，".，^q)+i k 1 _ 一 /S _ 1""-1 一 一 一 J 二 一 + (D3^k"(ej，Xi，...,VejXk,...，Xq)+ 5： (D3^^^)(i.,X,,...,VX,,...,X,,...,X)]} k+l n-l A ^^ A, = Ej= i (D,Dx^^)(i.,X^,…’ Xk，...，X^). Then by Ricci's formula, we have \ = Ak + ^1 + hl + A: + kl + Ak'，where n-l A A,i =E(D^D^^^)(H.,X^,...,X^,...,X^), n-l — A A,2 =5>(ej，^)(D#)(^，i^i”..，5k，...，^q)， j=l 4 n-l A Ak = X)K(Cj,v^)<^(ej,X^, ...’Xk，...，X)， j=i n-l 一 _ 二 < ="E^(R(ej,X,)e.,X^,...,X,...,X^), n-l 一 一 _ A Ak5 = E E ^(ej,X^,...,R(e.,X,)X,,...,X^,...,X), J=1 l^f First，we have f (-l^A,' = (\i^^ n-l 一 n-l jj_l ,^：(-!^,^ = j=ER(X,,epei . = -j:R(e.,X,)i.j=i . Hence k=i = -(R.^)(X,,...,X)q . Using Corollary 2.7.5. again, we have R(e.,X,)X, = R(e.,X,)X,, and it follows that n-l Ek = l (-l)k[Ak5+Ajf] = 0. Therefore, (1) is proved, Chapter 3: The Harmonic Theory ^j To establish (2), we use the Ricci formula and obtain E(DeD,v.)(X,,...,X^) j = i J J n-l 一 一 n-1 一 一 一 =E (�e,)(Xi”..’Xq)-； ^co(e.,ep(D^^)(X^,...,X) j=i j=i + ¾ K(ej，ej)—5^i，..jq)-2^W3^i，...，R(e^)Xk，...，X) j=i j=i k=i J J q n-l _ 一 =E (D-D^^^)(X,, ..., X^) - (n - l)iD^^ + R^ ^ , n-l n-l Since X) K(ej,ep = 0 and J^ o)(ej,Hp = -(n-l)i. j=i j=i (3) is obtained from (1) and (2) by eliminating the D#-terms and making use the operator Qq. • 3.2. The fundamental inequalities In the following, we shall further assume that the strongly pseudo-convex manifold M is compact. DEFEVITION 3.2.1. Define an inner product (，) in ^M, E) by (�^ )= < PROPOSITION 3.2.2. ^E is the formal adjoint operator of 毛，i.e. (¾^,…=( n-l |^||| = (Y) n-l (1) ||^||i = - (E (2) ||^||s = - (E n-l PROOF. It suffices to show that V { n - 1 n - 1 In fact, 5"a = Y. (V^a)(e ) = J] {ej - < Dg (D-^)， . Hence, by Green's Theorem, we obtain (1). (2) follows similar argument. • THEOREM 3.2.5. For any ip E C^(M, E), (1) (¾-, ^) = II 9 III - q(iD^ 9� +{K(p+R^ The following proposition is an intermediate consequence of the above theorem. PROPOSITION 3.2.6. Suppose, for some q, the self-adjoint operator Qj is positive definite at each x G M. Then W(M, E) = {0}. 3.3. Kohn's harmonic theory Denote the space of all smooth functions with compact support on R2n-i by QT(R2n-l). DEFB^ITION 3.3.1. For each s G M, define the Sobolev norm || ||“）ni QT(R2n-i) by l|f|l(s) = j(lH^I')Mf(OPd^ fGQT(R2n-l) A _ 2n-l where f(^) = (2ir) '^ e"'^^*^^f(x)dx is the Fourier transform of f. Let { Uk ’ hj^ }k e K be an atlas of M such that K is a finite set and such that each U^ is homeomorphic to R2n-i Let { Pj^} be a partition of unity subordinate to the atlas. For each k, take a moving frame {of, ...，e^.i} of S |^ and a moving frame {si^,…，Sr^} of E|u , where r = dinicE. We denote by P the family of all ordered set (i” …，i^) of integers with 1 < i^ < ... < i^ < n - 1. For any r II ^ ll(s) = E E E II (Pk- In the sequel, we fix an integer q with 1 < q < n — 2. DEFC^ITION 3.3.3. We define norms {| |{ and ||| ||| in C^(M, E) as follows: I (f ||2 = { II PROPOSITION 3.3.4. For PROOF. We obtain the result immediately from Theorem 3.2.5. since 1 < q < n - 2 and the operator Q^ is of order 0. • We shall make use of Kohn's inequality [FK] stated below: THEOREM 3.3.5. (Kohn's tiequality) Suppose Xj, ..., Xm are vector fields on the manifold M such that each X^，1 < j < m is a linear combination of the X:'s, and suppose that X：, [X. ,X.], [X. ’ [X;，X.]]，...， J ^1 ^2 M h l3 [X.，[... [X, , X. ] ... ] ] of order < j span all vector fields on M. Then if V is a relatively '1 'j-" ij compact subdomain of M, m II ^ ||(2-i) ^ C J2 II ^i^ V + ll\H|2 Li = l uniformly for all smooth functions x|/ supported in V，where C > 0 is a constant. Chapter 3: The Harmonic Theory ^j PROPOSITION 3.3.6. I 9 11(½) ^ C III PROOF. We first fix k. Let V be any compact subset of U^. Let C'o^(V) ={ ^ � 2n-2 2 广 k，\ z Y^ V k 2 k 7 r k k 一 - - 一 q _ 一 _ J2 (^x,^)li ^e = {D^cp)(Q. ^ --"^i,) = Xj(<;^(e., ...,ej^))-^ r (Dxi<>^)f,i = ^i Therefore, || cp \\^,^^ < C3 ||| cp |||，C^ is a constant. • The operator Dg is called subelliptic if there is a positive number 0 such that II By virtue of Theorem 3.3.7., we have THEOREM 3.3.8.网间 The operator Dg is hypoelliptic, i.e. (p E C^(M, E) is smooth wherever H^(p is. Following standard arguments, we have the main theorem in the harmonic theory. THEOREM 3.3.9. (Kohn) ^^K] (1) K^(M, E) is finite dimensional for any integer q with 1< q < n -2. (2) For each q with 1 < q < n - 2, there are unique operators H, G : (^(M，E) — C^(M, E) defined to be the orthogonal projection on Q^(M, E) and the inverse of D^ on the orthogonal complement of W(M, E) respectively such that (a) n^B = HG : 0，and (b) 0^0^? + R It is left to consider the cases where q = 0 or q = n - 1 (n > 3). We first define an operator H : P(M, E) — ^(M, E) by n PROPOSITION 3.3.10 [Ta]. The operator H defined above is an orthogonal projection of (?^(M, E) onto D^(M, E). Chapter 3: The Harmonic Theory ^j PROOF. Since q = 0，Dg = ^'^^. For any For q = n - 1，we defme an operator H : ^^'^(M, E) 一 C'^M, E) by U In the same way, we have the following proposition. PROPOSITION 3.3.11 [Ta]. The operator H defined above is an orthogonal projection of C^-i(M, E) onto iH^'^(M, E). We combine Theorem 3.3.9. with the above two propositions gives the following identification. PROPOSITION 3.3.12. Assume that n > 3 and let q be any integer with 0 < q < n - 1 • Then every cohomology class in H^(M,E) is represented by a unique harmonic form, and hence W(M, E) = HHM,E). 3.4. The harmonic theory and the duality Let M be a compact strongly pseudo-convex manifold of real dimension 2n - 1 and ^ a basic field on M. 3.4.1. DEFEVITION .• We define a Riemannian metric h on M by h = g + e\ Chapter 3: The Harmonic Theory ^j Thus, h(f,0 = 1, h(?,X) = 0, h(X,Y) = g(X,Y)forX,YEHp(M). The metric h induces a Hermitian inner product <，> in CT(M) defmed by DEFEVITION 3.4.2. Define an inner product (，) in ^(M) by { PROPOSITION 3.4.3. For any cp,xp G _/^(M), (cp, yp) = (n - 1)! (pAT"^. • PROOF. Bydefinition,((;i>, \^) = <^,;^>(n-l)! * 1 = (n-l)! Observe that there is a unique orthogonal decomposition CT(M)= CG(M) © S © S with respect to the induced inner product <，> . Hence we may A 一 identify the holomorphic tangent bundle T(M) = CT(M) / S with the subbundle CG(M) © S of CT(M), and hence CP,Q(M) = AP(t(M)*) ® A^S*. Recall that CP’q(M) is a subbundle of AP'q(M) and AP'q(M) = AP+q-UM) © CP,q(M). Let i^ be the projection of A^-^(M) onto CP'q(M) corresponding to the natural map y> ： AP,q(M) — C^(M,EP) defined in Proposition 2.3.5. Then, for cp E 0"^%M), the Chapter 3: The Harmonic Theory ^j tangential Cauchy-Riemann operator d” satisfies dj^ PROPOSITION 3.4.4. If DEFEVmON 3.4.5. Define the operator # : Ak(M) — A^^-^-^M) by ^ E A^M). The following two lemmas will be useful to prove a duality theorem on H^'^. LEMMA 3.4.6. For any positive integers p and q, 拉 C^'%M) = (^-p，n-q-i(M). PROOF. As discussed before, ^'^M) = 0 J^ B(r,s,q), r + s=p where B(r,s,q) = A^(CG(M)*) ® A'S* ® MS*. Moreover, ^B(r,s,q) = B(1 -r,n-l -s, n-l -q). Hence, # 0''%M) 二 � Y^ B(l-r,n-s-l,n-q-l) = (^"P'"-^"^(M). • r + s=p LEMMA 3.4.7. Let b" be the formal adjoint operator of ^ : (?’q(M) — (f^'q+i(M) with respect to the inner product ( , ). Then 5//0 = (-1” + 1样1#4 for X|y e (?»，q+i(M). Chapter 3: The Harmonic Theory ^j PROOF. Let ip E C'%M) and • E C^,Q+i(M). Then, by Proposition 3.4.4., d( (“…=<^M^'^>dv 亀 =(n-l)! I —d^ipfK^yp =(n-l)! d(pASi>^)+(n_l)!(_l)P+q+i ^K^fd^^yp J • =• =(。（-irq+i«ltw. Since b" is the formal adjoint operator of ~dy^, the result follows immediately. • Recall that E^ = AP(t(M)*) is a holomorphic vector bundle with respect to the differential operator 3gp in Proposition 2.3.4. It should be equipped with the inner product <，> as a subbundle of AP(M). By Proposition2.3.5., ^^,Q(M) = ^^(M, E?) and \ = (-l)P^p. Clearly, the inner product in (?(M, E^) induced from the inner product in E^ coincides with the inner product < ,> in (?，q(M). Hence we have 5" =(-l)P&" where ？？1^口 : 0^\M, E?) — (^%M, E^) is defined as in Definition 3.1.3. DEFESfITION 3.4.8. Define a differential operator A" by A" = b" \ + _d^b". Let W\M) 二、中 E <^,q(M) : A" PROOF. Since A"=�an W'\Md ) = 0^(M, E。，(1) is proved by Theorem 3.3.9. By Proposition 3.3.12.，we have (2). • By Lemmas 3.4.6. and 3.4.7., we have the duality theorem as below. THEOREM 3.4.10. Let p and q be any positive integers. #0 PROOF. Note that by Lemma 3.4.7., jtA" = A"S . Hence, for any h E lK P'q(M), A"h =0 implies that A"(#h) = 0. It follows that Sh G iKn-p'n-q-i(M). • Finally, we have the following. PROPOSITION 3.4.11. Suppose the Ricci operator R* is positive definite, i.e., the quadratic form g(R^X,X), X E Sp, is positive definite at each p G M. Then H°'^ = {0} for any q with 1 < q < n-2. PROOF. The result follows by applying Proposition 3.2.6. to the trivial holomorphic vector bundle M x C with the usual inner product. • Chapter 4 The Holomorphic Extension of CR Functions In this chapter, we give a proof of the holomorphic extension of a CR function by the analytic disc method. By definition, a function f: M — C is called a CR function if5j^f = 0 on M. It is well known that a holomorphic function restricted to a CR manifold is a CR function p5o]. However, not all CR functions are the restriction of holomorphic functions. An approximation theorem due to Baouendi and Treves \BT] is first introduced. By this theorem, a CR function on a embedded CR manifold can be locally approximated by a sequence of entire functions on C", Then by using an analytic disc theorem, the holomorphic extension result follows immediately. 4.1. Approximation theorem Suppose M is a generic embedded CR manifold of class C^ with din^ M = 2n - d， 0 < d < n. We choose suitable coordinates for such that p E M is the origin and M = {(z=x + iy,w)e C^ X C""^ : y=h(x,w)} where h: M^ X C""^ — M^ is of class C? with h(0) = and Dh(0) = 0. Let w = u + iv E C""^ and we introduce new variables (t, s) by t = (x,u) E Md X Mn-d 三 ^n ^nd s=(y,v) G R^ x R""^ = R". Now the coordinates for C" can be written as f = t + is. Define H : R" x M""^ — R^ x R""^ by H(t,v) = (h(x，u + iv)，v). ^TJ Since h(0) = and Dh(0) = 0，we have H(0) = 0 and _(0) = 0. We only consider a small neighbourhood of the origin and so we multiply h by a suitable cutoff function and assume that h has support in a neighbourhood Ui X U2 C Rn X W ‘ d of the origin. U! and U2 can be chosen small enough so that ^(t,v) I < 1 forall (t,v) G M" x R""^ 5t 2 5 H Here, ^ denotes the sup-norm of the matrix —. at ^t Chapter 4: The Holomorphic Extension of CR Functions joo By Mean Value Theorem, I H(t,,v)-H(t,,v) I < 1 I ti -t^ I for (tpV), (t,, V) E r x 肥、 Note that M is parameterized by H near the origin, i.e., M = {t + iH(t,v) : (t,v) G Ui X U2 C Rn x 肥-” M is also foliated near the origin by the n-dimensional slices Mv = {(t + iH(t,v) : tGU^ C r} for V G U^. LEMMA 4.1.1. Let g G OT(Ui) with g = 1 on a neighbourhood U/ with 0 G U/ C C U^ C R". Extend g to Cn so that g(f) is independent of Imf, for f E C". 门 ， Let f : M — C be a continuous function_ n . Then, for f E M^ { U/ x U^} f(n = lim7r"^fi-" g(r)f(r)e-r2[r"dr. “� Au^ Moreover, this lindt is uniform for v E U^ and f G M^ f| {U/ x U!}. PROOF. Define f : R" x Rn-d —� b"y r(t,v) = t + iH(t,v). For fixed v E U^ C 1“，M^ is parameterized by the map t' >^f(t^v) for t' E U^. Thus it suffices to prove that f(f(t,v)) = lim 7r"^£-" g(Of(r(t�v))e-r2[f(t’v)-f(t'’v)rdet(^_^(t/，v)ldt' "0 “R» [化 . where the limit is uniform in (t, v) E U/ x U:. Now we fix t and let t' = t - es. Then the above integral becomes 7r"^ g(t-es)f(f(t-es,v))e-r^t,v)-f(t-es,v)Pdet|^M(t-es,v)] ds (*) s“ pt If t E u/，then g(t -ss) — 1 as e — 0. Chapter 4: The Holomorphic Extension of CR Functions joo Moreover, f(t,v) 一 f(t-es,v) = -4^(t,v)-(ss)+0(e^). Hence, oi r2[f(t,v) - f(t-fis,v)f = ||(t,v)-s +0(e). Therefore, the integrand in (*) — • converges pointwisely in s (uniformly in t E U/ and v E U� a)s, e — 0，to f(f(t，v))exp{-[_y(t，v).s]2}detf4^(t，v). ot dt Note that g-f-—dt is globally bounded, Moreover | e-"[^'v)](t-es,v)]2 | = g-e-Re{[r(t.v)-f(t-^s,v)f}^ Wehave Re{[f(t,v)-f(t-es,v)f} = Re{[t+iH(t，v)-t+es-iH(t-es，v)]2} =Re{[es+i(H(t,v)-H(t-es,v))]2} =8^\ S |2-[H(t，V)-H(t-SS，V)]2 > e2| s P-lfi2 s 2 4 =le^ls P 4 Therefore, by the dominated convergence theorem, the integral in (*) converges to 7r'^f(r(t,v)) e-[#(t,v).s]2detf4i:(t,v)lds s4r at as e — 0 and the limit is uniform in (t,v) E U/ x U: C R" xR""^. ^ y 0 ^ Tjr Since (^)(t,vat ) = _(at t + iH(t,v)) = I + i-(tot , v),whereIisannx n identitymatrix, and recall that | 3——(t,v)H | < -1， the proof is complete if we show dt 2 _ n 7T ^ e_[A.s]2(detA)ds = 1 .....(**) s E R" for all matrices A G A ={n x n complex matrix A : | ImA | < - | ReA | and Re A is 2 non-singular}. Chapter 4: The Holomorphic Extension of CR Functions joo Consider the integral in (**) as a holomorphic function of the entries in A E A. Suppose ImA = 0. (**) follows by changing the variable s, = A .s G W and using a standard polar coordinate calculation. (**) must hold for all A E A by the identity theorem. • Note that the domain of integration M^ depends on f. In the next lemma, if f is CR, the domain of integration will be independent of f and we have an approximation theorem. LEMMA 4.1.2. Suppose f is a CR function of class C^ on a neighbourhood co^ of the origin in M. There is a neighbourhood co^ in M with 0 G o)^ C C co^ C M such that for f G c02 f(f) 二 lim7T_Srn g(r)f(r)e-r2[f-n2df/. 0。 •。 The limit is uniform in f E o)^. PROOF. Consider f:rxr-^^..C" by f(t,v) = t + iH(t,v) for some unique tE U” V G U^. Fix vGU2, let M^ = {f(t^Xv) = t' +iH(t:Xv) E M : t^ G U^ C M",0< X < 1}. Clearly, M^ is an (n + l)-dimensional submanifold of M and its boundary aSlv = MoUMvU{r(",Xv) : t' e aUi and 0 < X < 1}. Since g G Co" (U!), the only contributing components of dM^ are M^ and M^. Hence, by Lemma 4.1 • 1. and Stokes’ Theorem, f(f) = lim7r"^e-" g(^)f(t')^-'''^^-^^'dt' e- � r“ =lim7r'^e-" g(t')f(t)^~'''^^~^^'d^ e^0 “ J_ r € 3M,-Mo = limTT—h-n d,{g(r)f(r)e-r2[f-")df/+ g(r)f(f)e-r2[rn2dp> e — 0 , J — . ^ e M. r e M� Chapter 4: The Holomorphic Extension of CR Functions joo is the exterior derivative. Note that the only contributing terms in the integral over M^ comes from 5^. Assume that the CR function f can be extended ambiently to C" so that af = 0 on cOj C M • We choose U! and Uj in Lemma 4.1.1.small enough so that f{ U! x U^} C Wj. Hence • � f(f) = lim7r"^e-| (\(g(fO)f(r)e-r:[�"dr + g(r)f(f)e_r2[""dr,. "—0 r^M, rin, � (***) The result is obtained if the first limit in (***) is zero. Recall that f = t +iH(t, v). For ^ € M^, there is a unique X with 0 < X� suc1 h that f^ = f(t^Xv)=t' + iH(t',Xv). NowRe[r-n'= |t-t^P- |H(t,v)-H(t^Xv)|2. By ^TT 1 mean value theorem and |——（t,v)| S —， di 2 I H(t，v)-H(t\Xv) I < 1 |t-t^ I + C|v-Xv I 2 where C is some uniform constant depending only on the C^-norai of H. Hence, I H(t,v)-H(t^Xv) P < i_|t-tT+C2|v-Xv|2+C|t-t' v-Xv 4 < 1 t-tT+C2 V-XV 2+1 t-t^ 2+c2 v-Xv 2 4 4 < 1 t-t^ 2+2C2 V 2 2 Thus, Re[f-n^ > ||t-t'|2-2C2|v|2. As in Lemma 4.1.1., g (^) is independent of tof^, i.e.,g(f^) only depends on i' = Ref^. Since g s 1 on U(, 3^g = 0 on a neighbourhood of the origin. Thus, if ^ = t, + iH(t^Xv) E supp5^g, I t^ I must be bounded away from zero. Moreover, f = t + iH(t，v) — 0 if and only if t — 0 and v — 0. Thus, there are constantsr^, r^ > 0 such that if fGM^ with | f| < r^ and if r'E supp3^g,then Re[f_rT > r^. Chapter 4: The Holomorphic Extension of CR Functions joo Therefore, | £- (l(g(r))f(fOe_r2[f-n:df^ | < C£-e"^'^^' reu, for all f G Mv C M with | f | < x^. Here C is a uniform constant that is independent of e，v and f. Therefore, the first limit in (***) converges to 0 as £ — 0，provided f G Mv C M with I f I < r�.Th prooe f is complete. • An approximation of a CR function by a sequence of entire functions can be followed from the above two lemmas. THEOREM 4.1.3. Suppose M is a generic embedded CR manifold of class C^ with dinij^M = 2n-d, 0 < d < n. Given an open neighbourhood o)^ of p E M，there is an open set co^ C M with P G� C2 ^1 such that each CR function of class C^ on co^ can be uniformly approximated on 0^2 by a sequence of entire functions in C". 4.2. The technique of analytic discs To extend a CR function on an open set 0 C C", we first have the result that this CR function can be uniformly approximated on an open set co C M by a sequence of entire functions. This sequence of entire functions actually converge uniformly to a CR function on CO. For this, we use the technique of analytic discs. The idea is that each point in Q is contained in an analytic disc whose boundary image is contained in co. From the maximum principle for analytic functions, the sequence of entire functions also converge uniformly on Q. DEFEVITION 4.2.1. A cone T of M is a vector space defined by r = { V G C" : Xv E r, for all X > 0 }. Chapter 4: The Holomorphic Extension of CR Functions joo A wedge is an open set of the form w + { T 门 B^}，where + is the arithmetic sum, w C M is an open neighbourhood of p E M, F is a cone in the normal space Np(M) and B^ is a ball in Np(M) of radius e at p. Let D be a unit disc in C. An analytic disc is a continuous map A : D — C" which is holomorphic on D. The boundary of A is the restriction of A to the unit circle S^ = dD. Usually, we identify A with its images in C". We choose coordinates for C" such that p E M is the origin and M = {(z=x + iy,w) G C(X C" ： y=h(x,w)} where h: R^ x C""^ — R^ is of class C^ with h(0) = and Dh(0) 二 0. DEFEVITION 4.2.2. Let S^ be an unit circle in C. If u : S^ — R^ is a smooth function, then u is extended to a unique harmonic function on the unit disc D. This harmonic function has a unique harmonic conjugate in D，denoted by v, which vanishes at the origin. The Hilbert transform of u, Tu : S^ — R^ is defined to be v 1^,. If u + iv:D — Cd js an analytic disc and continuous up to S\ then T(u |5,) = V |5, -v(f = 0). Since -i(u + iv) = v - iu is also analytic,T(v |5,)= -u +u(f = 0). Given an analytic disc W : D — C"~^. We have to find an analytic disc G : D — Cd such that the boundary of the analytic disc A = (G，W) : D — C" is contained in M. Thus G must satisfy ImG(f) 二 h(ReG(f),W(f)) for | r| = 1. Now parameterize f by e'^, 0 < 0 < 2ir. Let G = u + iv. Then v(ei” = h(u(ei”，W(ei”). It implies that u - x = -T(h(u,W)), where x=u(f = 0). Chapter 4: The Holomorphic Extension of CR Functions §3 Hence, we have u(e'^) = -T(h(u(e'^),W(e'^))) + x. which is the Bishop's equation. Conversely, if u，v : S ^ — M are continuous functions with u = -Tv + x，x = u(i* = 0)，then u + iv:S^^C^ is the boundary values of a unique analytic disc G: D ^ C^ with ReG(f = 0) = X. We parameterize f by e'^, 0 < ^ < 2ir. The function 0 ^ u(e'^) + ih(u(e^^),W(e'^)) is the boundary values of a unique analytic disc G : D — C^. Clearly, dA C M as ReG(e'^) = u(e'^). Therefore, we have LEMMA 4.2.3. Let W : D — Cn - d be an analytic disc and x G R^. If u : S ^ — R^ is a continuous ftinction that satisfies u(ei” = -T(h(u(e>^),W(e'^))) +x, 0< We are going to give the solution of Bishop's equation. Note that the Hilbert transform is not a continuous linear map on the space of continuous functions on S^ with' sup-norm. Thus, we consider the space of Holder continuous functions. DEFEVITION 4.2.4. Suppose L is any normed linear space. A continuous ftinction f : S^ — L is Holder continuous with exponent a (0 < a < 1) if there exists a finite number M > 0 such that |f(e^^') -f(e'^0| Note that C"(SO is a Banach space under the norm ||fL=|fL+ sup |f(ei�-f(ei�l o^0,,0,^2T 0j -2 r where | f | ^ is the usual sup-norm. PROPOSITION 4.2.5. The Hilbert transform is a continuous map from C"(Si，R^) to itself. PROOF. It suffices to consider the case d = 1. If u : S1 — R is continuous, its harmonic extension is given by Poisson's integral formula u(rei, = +2fT^i^i^^^ 0化1，0化2,. 27T .j 1 -2rcos(0 -6) +r^ Let z = rei6, f = e'^. Then &6 = J-df and we have iz u(z) = A i^u(n^ 2兀1 ir1=i I f-z |2 r ‘ 飞 = Re ^ f^u(D^ ‘ � 27Tl|"irZ f一 Since -；1 ^ ^~~u(ft + z ) _d^ ^ is holomorphic in z for | z < 1，a harmonic conjugate of 2^1 |fj.i r-z f ^ u is given by ‘ � v(z) =Im ^ pu(n^ '. 2n!r=if-z f 、 一 Chapter 4: The Holomorphic Extension of CR Functions joo _ � 2v Note that v(0) = Im ‘ ^ u(e'^)d0 ‘ = 0. Therefore, Tu is the boundary value of vonS� The function (广广)is smooth for | f | : 1 and | z | < 1. Therefore, the continuity of T will follow from the continuity of the Cauchy kemel K : C"(SO — C"(SO, where Ku is the boundary values on S^ of the function (Ku)(z) = ^ ^df, |z| < 1. 27T1 |丄1 f-Z To show the continuity of K on C"(SO, we first extend u to a function E(u): D — R such that II E II c«(5) ^ C II u ||c"(s') for some uniform constant C that is independent ofu. To construct E(u), we extend u to be constant on any line normal to S\ Then we multiply this function by a suitable cutoff function. Hence, it suffices to show K : C"(D) — C"(S^) is continuous. For z E D, Cauchy's integral formula gives Ku(z) = 1 u(q-u(z)dh(z) 2^1 |fj=i f-z Thus, |Ku(z)| _ _ _ 1 u(f)-u(Zi) u(f) -u(Z2) ^^ — — Q r 2�JmA [ Hi "2 JJ + 丄 u(z^) -u(z^) ^reJ-B.L f� u(f)-u(Zi) u(f)-u(Zi)"| ,, + a r f-A f-Z2 _ + u(Zi) -u(Z2). By Cauchy's integral formula, 1 u(Z2)-u(Z1) 1 u(Z1)-u(Z2) 171 ~~T^^~dr = B _•—_dr + u(Z2)-u(Z1). fes''-B,, f ^ ^^^6^'nB,. ^ ^ Hence, K(u)(Zj) - K(u)(、）= \ + A^ + A3, where • »1 1 u(n-u(Zi) u(f)-u(z2) ^, A, = i_ 一 i_ d^ 2� U rzi f-z, A. = ^ u(zfu(Z2)dr,and 2��nB2 ,^-¾ 丄 (Zi-Z2)(U(r)-U(Zi)) �^,J.-B. . (r-z,)(r-z,)[ Notethat|Ai| < ^ (| f-z^ |"-^ + | f-z^ |""0|dr| • || u ||^. Parameterize Zifes^'ABi, z 'e the unit circle by f = _L_e“. Then | A^ | < C t""Mt-|| u ||^, where C is a uniform Zi “ constant. Since e = | z^ - z^ | and a. > 0，| A^ | < C^ || u || ^. | z^ - z^ | “. Similarly, we have | A3 | < C^ || u ||^-| z^ -¾ |". Chapter 4: The Holomorphic Extension of CR Functions joo Using Cauchy's theorem, A^ = ^ u(z^) ^(¾)^^ for z,，z，E D. 2�ife‘r D ^-¾ For fE 哗"I f-Z2 I > £. Hence, IM<^|Zi-Z2l°!u|L.2w = h|la.|Zi_y„ Thus, |K(u)(Zi)-K(u)(z^)l^:||u|| 1¾-¾!" where C^ is a constant depending only on a.. This estimate is uniform for z^，� DG. Therefore, this estimate is uniform for z^ ’ z^ E D. The proof is complete. • For u:Si — M, W:Si — C^-\ define H(u,W) : S^ — R^ by H(u,W)(e'^) = h(u(e'^),W(e'^)) for 0 < 0 < 2ir. Suppose h is of class C^ and let 0 < a <1. H : C"(Si, M^) x C"(Si, €："-” — 0：«(51，R^) is then a continuous linear map. If h is of class C:，then H is a C^ map in the sense of Banach spaces. And we have (D^H)(u,W)(v) = |^^(u,W)-v. dx Here, D^^H is the Banach space derivative with respect to u which is defined for u，vGC«(Si，Rd)by (D^H)(u,W)(v) = i;mH(u+tv，W)-H(u，W) t—o t the limit is taken with respect to the topology on the space C"(S ^，R^). Since T : C«(S\R^) — C"(Si,Rd) is a continuous linear map, T is also differentiable and (DT)(u)(v) = T(v) for u,vG C^(S',R'). We give the solution to Bishop's equation as follows. Chapter 4: The Holomorphic Extension of CR Functions S8 THEOREM 4.2.6. Fix 0 < a < 1. Suppose h : R^ x C""^ — R^ is of class C^ with h(0) = 0 and Dh(0) = 0. There is a b > 0 such that if | x | < 5 with W E qSi，C""^) with I W II a < 6，then there exists a unique u : u(W，x) E C"(S ^，R^) that satisfies Bishop's equation with u(0,0) = 0. If h is of class C^ (k > 2), then there exists a h > 0 such that . u:C«(Si，Cn-d)xRd — c«(Si’Rd)depends on a C^"^ fashion on xEr and W G C°^(Si，Cn-d) with I X I < 5，|| W ||^ < 6. PROOF. Define F:C«(Si，Rd) xC«(Si，Cn-d) x M^ — C"(SM^) by F(u,W,x) = u + T(H(u,W))-x. It is of class Ci in a neighbourhood of the origin in C"(S\ R^) x C"(Si, C""^) x R^. Atu = 0，W = 0, X = 0，sinceDH(0) = 0，D^F(0,0,0)(v)=v. Hence,D^F(0,0,0) is the identity map I. Since F(0,0,0) = 0，by implicit function theorem,u = u(W，x) exists. ,.. Since F is of class C\ u depends on a C^ fashion on W and x in a neighbourhood of origin in C"(Si,Rd) X Rd. If h is of class C\ k > 2, then F is of class C^ ^ By induction, u depends in a C^'^ fashion on X and W in a neighbourhood of the origin. In fact, D^F(0，0，0) = I. Differentiating F (u，W，X) = 0 allows us to solve for a jth derivative of u in terms of lower order derivative of u. • DEFEVITION 4.2.7. Suppose r^ and r^ are two cones in the normal space Np(M), p G M. 1\ is said to be smaller than r^, denoted by r^ < r^, if r^fjU is a compact subset of the interior of r*2 门 U，where U is the unit sphere in Np(M). Chapter 4: The Holomorphic Extension of CR Functions joo For p E M, let T^ = { the convex hull of the image of ^} C Np(M). Clearly, Fp is a cone. We choose coordinates so that p is the origin and M = {(z=x + iy,w) G C^ X C""^ : y=h(x,w)} where h:R^ X C"-^^R^ is of class C^ with h(0) = 0，Dh(0) = 0 and jm=0 for|a|+m=2. dx"3w^ Identify Ho'^M) with {(0,0,w) : w E C""^} and N^(M) with {(0,y,0) : y G M^}. Define a map S : e-~> ^(w) and it can be identified with the map w ^ X (w,w) G R^. Clearly, r� ithse convex hull of the image of this map. Suppose r < r� Le. t T^ ^, be a convex cone generated by vectorsX^，..., X^ E N^CM) such that r < r^I x^ - ^o- Since X^ lies in the image of ^, there exist vectors ojj G cn-d with Xj = ^(aj,ap, 1 N define the analytic disc WGC«(Si，Cn-d) by W(t,w)(f) = w+J^ t.a.^^ with i = l w(o,o)(n = 0. ByTheorem4.2.6., u(t,x,w)= u(W(t,w),x) is the unique solution to the Bishop's equation u + T(H(u, w)) - x = 0. By Lemma 4.2.3., the analytic disc A = (G, W) has its boundary contained in M where the analytic disc G satisfies (1) and (2) in Lemma 4.2.3. Moreover, since h is of class C^, u is of class C^ and hence A is of class C^ The uniqueness of the solution to the Bishop's equation implies that u(0,0,0) = 0 and hence A(0,0,0) = 0. We summarize all these in the following lemma. Chapter 4: The Holomorphic Extension of CR Functions 90 LEMMA 4.2.8. Let 0"(D, C") be the set of all analytic discs with values in C" whose boundary are elements of C «(S ^，C"). Given an open neighbourhood Wj of the origin in M. There is a 5 > 0 and a map A : { (t,x,w) G R^ x R^ X C""^ : |t|, |x|，|w| < 5} — 0"(D, C") of class C3 such that the boundary of each A C co^ for 111，| x |，| w | < 8. Moreover, A(t,x,w)(f) = (G(t,x,w)(r),W(t,x,w)(f))andReG(t,x,w)(f = 0) =x. Next, we have to show that the set of centres fl = {A(t,x,w)(r = 0) : |t|, |x|, |w| <8} contain a set of the form co^ + {T 门 BJ for some e = s^, > 0. Identify (z, w) with (x, y, w) where z = x + iy. LEMMA 4.2.9. Given rj >0, there are d > 0’ e' > 0 such that for | x |，| w | < 5 and 111 < e'， N A(t,x,w)(r = 0) = (x,h(x,w),w) + (0,j]t/x.,0) + (0，^(t，x，w)，0) 、八 1. =. 1 ^ where ^ : R^ x R^ x C""^ — R^ is of class C^ and | ^(t,x,w) | < rj | t p. PROOF. We write G (t,x,w)(r) 二 u(t，x，w)(r) + iv(t,x,w)(r). By Lemma 4.2.8., W(t,w)(0) = w and u(t,x,w)(0) = x. Hence, A(t,x,w)(0) = (G(t,x,w)(0),W(t,x,w)(0)) =(x,v(t,x,w)(0),w) We are going to examine the Taylor's expansion of A(t,x,w)(0) in t for x, w fixed with X I，I w I < 5. Since v(t,x,w)(f) is harmonic in f, the mean value theorem gives lr v(t,x,w)(0) = ^2 “ v(t,x,w)(e'^)d0. Since bK C M, we have v(t,x,w)(e'^) =h(u(t,x,w)(e'^),W(t,w)(e'^)). 2r Thus, v(t,x,w)(0) = ^ h(u(t，x，w)(ei”’W(t，w)(ei”)d^. ..…(*) 2 7T丨 Chapter 4: The Holomorphic Extension of CR Functions joo Ift = 0，W(0,w)(f) = w which is a constant, then u(0,x,w)(e^^) = x is the unique solution to the Bishop's equation. Hence, 2x v(0,x,w)(0) = ^ h(x,w)d0 = h(x,w). 27T .丨 Differentiate (*) with respect to t at t = 0，we have ^:(0，x,w)(0) = lT^(x,w).|^(e^^)d0 + atj 2ir dx dt. t=o (氺氺) • l2Re|^(x’w).^(e”[|t�. Q u Since u = ReG, ^(0,x,w)(f) is harmonic in f for |f| o u(t,x,w)(0) = X, hence, ^(t,x,w)(0) = 0 for any t. By mean value theorem, ^h 5 27T _(0,x,w)(0) = — ^(0,x,w)(e'^)dcA = 0, and so the 1st integral in (**) vanishes. J T) J Similarly, the 2nd integral in (**) vanishes since ^(t,w)(0) = 0 atj Therefore, we have dv ^(0,x,w)(0atj ) = 0 1 ‘ � Now, ^(x，：) = d'h(0,0) + a^h(x,w) - a^(0,0) ^w^aw^ aw^aw^ dw^aw^ dwjw^ • The 2nd and 3rd terms can be made as small as desired by suitably restricting x and w. Therefore, V^^(0,x,w)(0) = gp[^(e-),^] d, + yt，x，w). where | ^.^(t,x,w) | < ” | t |^ for (t,x,w) is in suitably small neighbourhood of the origin. 3 'Yy Since _dt j = a-^, by putting f = e'^ 0 < < 27r, we have 广 、 27T 1 •笔 g(ei”，^^ d^=Oforj#k. Pj at/� 产 � 27T Ifj=k, S ^、己'”,仏 d(^> = X.. Hence, ^S dt, J j •- tt d^v l^(t,x,w), forj ?f k, J^^(0,x,w)(0)= 2 dtjd � [t/X.^|..(t,x,w), forj=k. Since the 3rd-order Taylor remainder in t can be absorbed into the error term, the proof is then complete. • Chapter 4: The Holomorphic Extension of CR Functions joo We shall show that, for fixed x, w with | x |，| w | < b，the map N t-f(x,w)(t) = j Y= .i t/x +f(t,x,w), tERN , |t| N Now t ^ Y^ tj2 Xj，111 < e' parameterizes an open neighbourhood of the origin of the cone j = i ^x,，".’X, > r. We claim that since | f(t,x,w) | is small relative to | t「，the image of t ^ f(x,w)(t) will also contain the desired neighbourhood of r. First, we replace t. by s|^, for t^ > 0. Then we define A N F(ti,...,tj^) = ^ t.X. and j = i F(x,w)(ti，."，tN) = f(x,w)(y^^,...,y^) = f(ti，...，tN)+e(t，x，w) where the error term e(t,x,w) = Hf~” ..., y^^,x,w) is continuous with e(t,x,w) I < r] I t I , provided |x|，|w| < d and t < lf^. N Here, rj > 0 can be chosen as small as desired and 8 depends only on rj. We wish to show that d can be chosen so that if | x |，| w | < d, then the image of the set /p/ )2 {t E R^ : t. > 0，111 < i^ } contains an s-neighbourhood of the origin of r. Since r < rx,,...,XN, for each v G fflU, where U is the unit sphere in R^ there is a conical neighbourhood of v, denoted r^, and a collection of d-linearly independent vectors X，... X with V G Fv < r^ …，x . Since f 门 U is a compact subset of r^ ^，we can ‘ “ *| M ^J , • • • > A,JyJ cover r by a finite number of such r^. Therefore, it suffices to show the following lemma. Chapter 4: The Holomorphic Extension of CR Functions joo LEMMA 4.2.10. Suppose Xj, ...,Xd are linearly independent vectors in R^. Suppose F < r^ ^ and e' > 0 are given. Then there exist rj，e > 0 such that if F : R^ — R^ is a continuous map with I F(t) -F(t) I < 7}\i\ fort = (t^, ...,t^) witht� 0>, 0 PROOF. Assume that {X^, ...，XJ generates R^ as standard basis vectors. Hence, F is the identity map and r^ ^ ^, “ { (Xi，...，x^) E R^ : x^ > 0，1 < j < d}. Given r < Fx, Xd，there is a V > 0 and 0 < e < e^ such that if x G r 门民，then the Euclidean distance from x to d { Fx x,门 B^'} is greater than rj^ | x . Suppose I t - F(t) I < rj | t | . If rj，e > 0 are chosen small relative to r\' and e,, then the line segment between t E 5{ [乂乂门民,} and F(t) does not intersect rAB^ (Figure 1). Suppose X E r f| B^ is not in the image of r^ x, D By under F. The restriction of F and the identity map to a{rx x< -¾^ r imageof X / \ ^^ a{rx...^nB,} /^ \^^ :^^ts' S i ‘ Figure 1 Chapter 4: The Holomorphic Extension of CR Functions joo On the other hand, T^ x fl ^e' ^s contractible to the origin in R^. Since F is continuous and X is not in the image of F, the image of d{T^^ ^, H B^/} under F is also contractible in Rd - {x}. Hence, the homology class of F{ d{T^ ^,门 B^/} } is trivial in the (d - l)st dimensional homology group of R^ - {x}. This contradiction proves the lemma. • By Lemmas 4.2.9. and 4.2.10, we have the following result concerning analytic discs. THEOREM 4.2.11. (Analytic Discs) Suppose M is a generic embedded CR manifold of class C^, 4 < k < oo with dim^M = 2n — d, 1 < d < n - 1. Let p G M such that the interior of T^ in Np(M) is non-empty. Then for each neighbourhood co^ of p E M and for each cone T < Fp, there is a neighbourhood cOp C o)^ and a positive number e^ such that each pointcop + {rf|B^ } is contained in the image of an analytic disc whose boundary image is contained in o)^. 4.3. Holomorphic extension We now come to the main theorem of the holomorphic extension of CR functions. THEOREM 4.3.1. (Holomorphic Extension) Suppose M is a generic embedded CR manifold of class C^, 4 < k < oo with dinijjM = 2n -d, 1 < d < n - 1. Let p E M such that T^ has non-empty interior with respect to Np(M). Then for every neighbourhood A of p G M, there exists an open set CO, C M and open set Q E C" such that (1) pGo/cSnMCA， (2) for each open cone Fj < T^，there is a connected neighbourhood co^ of p in M and an s > 0 such that cj^ + { r^ 门 B^} C Q，and (3) for each CR function f of class C^ on A C M，there is a unique holomorphic function F defined on Q and continuous on Q U �'suc thah t F |。，= f. Chapter 4: The Holomorphic Extension of CR Functions joo ^ A ‘ M '\ ‘ Figure 2 In Figure 2, the picture on the right is a side view with M going into the page. I"i f| Bg is represented by the shaded region. Clearly co^ and e depends on T^. The closer Fj gets to Fp，the smaller o)^ and s. Note from (2) that the tangent cone of Q at p is at least as large as Tp(M) + T^. If T^ = Np(M)，we may let r^ = Np(M) because r^ < T^. In this case, Q contains an open set {co + B^} which is an open neighbourhood of p in C". Hence, each CR function near p is locally the restriction of a holomorphic function defined in a neighbourhood of p. PROOF OF THEOREM 4.3.1. (Existence) Let p G A C M and let fbe a CR function on the open set A. There is a sequence of entire functions {F^} which uniformly converges to f on some open set ⑴之 with p G co^ C A C M. We apply the analytic disc theorem. Let I"p be a cone at p. There is a neighbourhood Wp C A and Sp > 0 such that each point in cop + {r 门 Bg } is the image of an analytic disc D and the boundary image of D is in A. LeuniformlClearlysubset Qt o=, f y pr 0 U (Uniqueness) Let co^ be a neighbourhood of p E M and e > 0. There are a e^ > 0 and a neighbourhood w^ of p with 0^^ C co! such that if F is holomorphic on w^ + {T^ 门 BJ and continuous up to co^，then F is the uniform limit on co^ + {T^ 门 B^ } of a sequence of entire functions { F. }. Suppose F = 0 on co^. Assume that the sequence of entire functions { F.} converges uniformly to zero on 0^. By the analytic disc theorem, there is an open subset U of ^2 + {^1 n ^e} such that each point in U is contained in the image of an analytic disc whose boundary is in o)^. By the maximum principle, F�converge tos zero at each point in U. Hence, F = 0 on U. Finally, by the identity theorem, F = 0 on co^ + {r^ ABJ. • EXAMPLE 4.3.2. Suppose that M is a quadric submanifold of C^ with real codimension 2 as defined in Chapter 1. We have shown in Theorem 1.3.8. that M is biholomorphic to one of the following four normal forms. (i) M = {(Zj,Z2,W1,W2) G C^:yi = q1(w,w),y2=O,wG C2} where q^ is a scalar-valued quadratic form. (ii) M = {(Zi，Z2’Wi，W2)eC4:yi — Wi|2，y2=|W2|2} (iii) M = {(Z1,Z2,W1,W2) e C4:yi = | w�p,y2=Re(WiW} (iv) M = {(Z1,Z2,W1,W2) G C^:y1=Re(W1W2),y2=Im(W1W2)}. For (i), the Levi form of M at the origin is contained in the line {(y, 0) : y^ E R }. The interior of the cone r�wit respech t to No(M) = R^ jg empty. Hence, the holomorphic extension theorem does not apply. For (ii), the image of the Levi form is the closed quadrant {y^ > O,y2 > 0} which is convex. Let r^ = {y^ > O,y2 > 0}. There is a cone r^ < r� N^CMC ) = 1^ and B^，s > 0 depending on r^，such that a CR function extends to a holomorphic function on an open set 0 G C^ containing A + {B^ 门 1\ }. Chapter 4: The Holomorphic Extension of CR Functions joo yw ~"""^{y|^ 八 / \ {y>=|w.in I y, _L^e Figure 3 Note that M is the intersection of the convex boundaries {y^ = | w^ |^} and {y2 = I w^ P}. Hence, CR functions on M cannot extend holomorphically past {Yi ^ I Wi P,y^ > I W2 |2} (Figure 3). For (iii), the image of the Levi form is {y^ > 0 } together with the origin. It is convex but not close. Then any CR functions on co holomorphically extend to an open set Q C C4 whose normal cross section at the origin contains sets of the type B^ 门 r^ where r*i < r� ={yj > 0}. The tangent cone of Q at the origin contains the half space {Yi>0}. For (iv), the image of the Levi form is all of No(M) = R^. Therefore, any CR function in a neighbourhood of the origin in M is the restriction of a holomorphic function defined on a neighbourhood of the origin in C^. EXAMPLE 4.3.3. We shall show Hans Lewy's holomorphic extension theorem 间 for hypersurfaces by using the technique of analytic discs. The theorem states as follows: Let M = { z G C" : p(z) 二 0 } be a hypersurface in C", n > 2, ofclass C^, 3 < k < 00， where p : C" — M is smooth with dp 5^ 0 on M. Let p G M and | Vp(p) | = 1. Let ir = {zE Cn ： p(z) > 0} and Q' = {zG C" : p(z) < 0}. (1) If the Levi form of M at p has at least one positive eigenvalue, then for each open set 0) C M with p G co，there is an open set U E C" with p G U such that for each CR function f of class C^ on co, there is a unique holomorphic function F on U 门 Q +, continuous on UAQ+ such that F |^pj^ = f. Chapter 4: The Holomorphic Extension of CR Functions joo (2) If the Levi form of M at p has at least one negative eigenvalue, then the result of (1) holds with Q + replaced by Q “. (3) If the Levi form of M at p has eigenvalues of opposite sign, then for each open set A C M with p G A, there is an open set U G C" with p E U such that each CR function of class C^ on A is the restriction on U 门 A of a unique holomorphic function defined on U. Recall that the Levi form can be identified with the map (Wi，...，w„_i)4g fM^W^W,. j,k = i dw.dw^ We may choose coordinates so that p G M the origin and M = {(z=x + iy,w) G C X C"] ： y=h(x,w)} where h : R x C""^ — M is of class C^ and all the 2nd-order pure terms in the expansion of h about the origin vanish. The Levi form of M then describes the 2nd-order concavity of M near the origin. A positive eigenvalue of the Levi form shows that M is locally concave upward in one of the w-directions. In this case, CR functions holomorphically extend above M. Similarly, a negative eigenvalue of the Levi form leads to a holomorphic extension of a CR function below M. If the Levi form has eigenvalues of opposite sign, then the origin is a saddle point for M and the holomorphic extension of CR functions to both sides of M is possible. n-l Now, the Taylor expansion of h about the origin is h(x,w) = ^ q.j^w^w^ + 0(3)， j.k = l where (aJ = ^ h(? is the matrix of the Levi form of M at the origin and dWj3Wk ]邮“ 0(3) are the terms involving both w and x which vanish to order 3 at the origin. Let Q = (qjj). Since Q is the Hermitian symmetric matrix, we can find a unitary matrix U so that by employing the transformation w = U.w，Q is diagonalized to U^QU. Let Q be a neighbourhood of the origin in C". Then M divides Q into two sets 0' = {(z,w) G Q : y > h(x,w)} and Q" = {(z,w) G Q : y < h(x,w)}. Chapter 4: The Holomorphic Extension of CR Functions joo y n+ 、， Do>yA-.0 // y _X,_/…人 ^ ^/^ �A-~«0 —=^^ > |w,| Figure 4 Suppose Q has at least one positive eigenvalue. This corresponds to the case where { y > 0 } C r� B.y reordering the w-coordinates if necessary, we assume q^ > 0. Then h(0,Wi,0) = q!i I Wi |2 + 0( I Wj |3). Let A be an open subset of M which contains the origin. Since q" > 0，any small translation of the complex line{(0,Wi,0) : w^ E C} in the positive y-direction will intersect Q^ in a simply connected open subset of this translated complex line whose boundary is in A (Figure 4). By continuity, this can be done in x, W2, ...，w^_i directions. More precisely, there are 5，e� suc0 h that if | y | < e and | x |，| w^ |，…，| w^_^ | < 6，then the complex line Ax.y.w,,.•”w„_| = { (x + iy，f，W2，."，w„_i) : ^E €} intersects Q + in a simply connected open subset D^ ^ ^ ^ of A^ ^ ^ ^ and its boundary A,>,Wj,...,W^^j A,y,Wj,...,W^_j J isin A. For | y | < e, |x| , |wJ，…’ | w^_^ | < 5，theunionofthe D� contains x,y,Wj,...,Wn-i an open set 0 + of Q +. By Riemann mapping theorem, each D�w2，...， w„_is, biholomorphic to a unit disc in C. This is the desired analytic disc. Consider q" < 0. This corresponds to the case where r^ = { y < 0 }. Then the same argument can be used to construct an open set Q - C n ‘ which is foliated by the images of analytic discs whose boundaries are contained in A. Ifthe matrix Q has eigenvalues of opposite sign, then Q+ U ^" U {『门乂} forms a set 0 in C" which contains an open neighbourhood of the origin and which is foliated by analytic discs whose boundary is in A. Bibliography [BT] M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. Math. 113， 387 - 421 (1981). [Bi] E. Bishop, Differential manifolds in complex Euclidean space. Duke Math. J. 32, 1 - 22 (1965) [B1] D. E. 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