Separate holomorphic extension of CR functions

We consider a real analytic foliation of by complex analytic manifolds of dimension m issued transversally from a CR generic submanifold of codimension m. We prove that a continuous CR function f on M which has separate holomorphic extension along each leaf, is holomorphic. When the leaves are cartesian straight planes, separate holomorphic extension along suitable selections of these planes suffices and f turns out to be holomorphic in a neighbourhood of their union. If M is a hypersurface we can also specify the side of the extension, regardless the leaves are straight or not.     

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

We consider holomorphic mappings sending a given Levi-nondegenerate pseudoconcave hypersurface M in Cn+1 into a nondegenerate hyperquadric of the same signature in PCN+1 and show that if M is sufficiently close to a hyperquadric in a certain sense, then any two such mappings differ only by an automorphism of the hyperquadric.     

An edge-of-the-wedge theorem for hypersurface CR functions

The classical edge-of-the-wedge theorem for holomorphic functions is generally false for CR functions. However, it is true on Levi-indefinite hypersurfaces for wedges pointing in null directions.     

Determination of formal CR mappings by a finite jet

We prove that if M is a connected real-analytic holomorphically nondegenerate hypersurface in Cn+1, then for any point p∈M there exists an integer k such that any two germs at p of local biholomorphic mappings that send M into itself and whose k-jets agree at p are identical.The above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose Jacobian determinant does not vanish identically.     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

Removable singularities of CR functions on singular boundaries

The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature of singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. Received: 8 December 2000; in final form: 24 June 2001/Published online: 1 February 2002     

The singular integral equation on a closed piecewise smooth manifold inCn

In this paper, we obtain the general permutation formulas and composition formulas of singular integral of the Bochner-Martinelli type on a closed piecewiseC ⁽¹⁾ smooth manifold. As an application, we consider the corresponding singular integral equation of linear variable coefficients, and prove that the singular integral equation can be transformed to an equivalent Fredholm equation, whose characteristic equation has a unique solution in , here denotes the function set which satisfies the Hölder condition on D and holomorphically expands to domainD.Corresponding author. Project supported in part by the Mathematical Tian Yuan Foundation of China (Grant No. TY10126033).     

On functions with one‐dimensional holomorphic extension property in circular domains

In this paper we consider continuous functions given on the boundary of a circular bounded domain D in , , and having the one‐dimensional holomorphic extension property along family of complex lines, passing through a finite number of points of D. We study the problem of existence of holomorphic extension of such functions into D.     

Analysis of the Kohn Laplacian on quadratic CR manifolds

We study the Kohn Laplacian □_b^(q) acting on (0,q)-forms on quadratic CR manifolds. We characterize the operators □_b^(q) that are locally solvable and hypoelliptic, respectively, in terms of the signatures of the scalar components of the Levi form.     

Characterisation of homogeneous polynomials which are constant on complex-tangential curves in the boundary of the unit ball of

We prove that a complex-tangential curve γ in the boundary of the unit ball of having the property that there exists a homogeneous polynomial P such that P=1 on γ has constant curvature. This implies that a homogeneous polynomial P having the property that there exists a closed complex-tangential curve γ (respectively a totally real 2-dimensional submanifold) in the boundary of the unit ball of such that P=1 on γ (respectively |P|=1 on γ) reduces to a monomial by a unitary chage of variables. These results represent a positive answer to conjectures of H. O. Kim.     

Logarithmic capacity and Bergman functions

Upper and lower estimates for the Bergman kernel and the Bergman metric in dimension one are given in terms of the logarithmic capacity; some of their applications are added.The second named author started the work on the paper while he was a guest at the Max Planck Institut für Mathematik in Bonn, Germany.The research was partially supported by the KBN grant No. 5 P03A 033 21 and by the Niedersächsisches Ministerium für Wissenschaften und Kunst, AZ. 15.3-50 113(55) PL.     

The Plemelj Formula of Higher Order Partial Derivatives of the Bochner-Martinelli Type Integral

In this paper, by using the technique of integral transformation, we obtain the Plemelj formulas with the Cauchy principal value and the Hadamard principal value of mixed higher order partial derivatives for integral of the Bochner-Martinelli type on a closed smooth manifold ∂D in Cⁿ. From the Plemelj formulas and using the theory of complex partial differential equation, we prove that the problem of higher order boundary value D^κΦ⁺(t) = D^κΦ⁻(t) + f(t) is equivalent to a complex linear higher order partial differential equation. Moreover, given a proper condition of the Cauchy boundary value problem, the problem of higher order boundary value has a unique branch complex harmonic solution satisfying Φ⁻(∞) = 0 in Cⁿ\∂D.     

On the characterization of additive functions with quadratic arguments

Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

Monodromies at infinity of confluent A-hypergeometric functions

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson [2]. In particular, we extend the result of [38] for non-confluent A-hypergeometric functions to the confluent case. The integral representation by rapid decay homology cycles proved in [9] will play a central role in the proof.