On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

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.     

Algebraicity of real analytic hypersurfaces and blowing-down

We show that a germ (M, 0) of real analytic strongly pseudoconvex hypersurface is holomorphically equivalent to a germ of real algebraic hypersurface if and only if there is a blowing-down from (M, 0) to a germ of real algebraic hypersurface.     

Compact Heisenberg manifolds as CR manifolds

Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup, with the natural strongly pseudoconvex CR structure. We identify the eigenvalues and eigenforms of the Kohn Laplacians on M and show how to realize M as the boundary of a bounded domain in a line bundle over an Abelian variety.     

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ ^N and a Levinondegenerate hypersurface in ℂ ^N+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].     

Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group

We present a characterization of the open unit ball in a separable infinite dimensional Hilbert space by the property of automorphism orbits among the domains that are not necessarily bounded. This generalizes the recent work of Kim and Krantz [6]. Key new features of this article include: a lower bound estimate of the Kobayashi metric and distance near a pluri-subharmonic peak boundary point of the domains in Banach spaces, an effective localization argument, and an improvement of weak-type convergence of sequences of biholomorphic mappings of domains in Banach spaces.     

The Ribaucour transformation for flat Lagrangian submanifolds

We study the Ribaucour transformation for flat Lagrangian submanifolds in complex flat space and complex projective space. As a consequence, we obtain a process to generate a new family of such submanifolds from a given one. Analytically, this provides a method to construct new solutions of the corresponding systems of PDEs from a given one. We also show that such transformation always comes with a permutability formula.     

Sharp lipschitz estimates for operator $$\bar \partial _M $$ on a q-pseudoconcave CR manifoldon a q-pseudoconcave CR manifold

We construct integral operators R_r and H_r on a regular q-pseudoconcave CR manifoldM such that

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Large classes of minimally supported frequency wavelets of<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ) and<Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>(ℝ)

We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H ² (ℝ)and symmetric MSF wavelets of L ² (ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets of L ² (ℝ)and H ² (ℝ).We also enumerate some of the symmetric wavelet sets of L ² (ℝ)and all wavelet sets of H ² (ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L ₂ (ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin.     

A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

The convex-concave principle and uniqueness for hypersurfaces in space Forms

We establish very general Weyl identities for pairs of symmetric functions of the invariants of the shape operator of a hypersurface in a space form or a general Codazzi tensor, respectively. They are used to characterize certain isoparametric hypersurfaces by assuming constancy or extremal properties of the functions, provided they fulfill ellipticity and/or convexity (concavity) properties. This way, many wellknown results are generalized. Finally, a chain rule for Weyl identities offers additional extension of some results.     

Normal analytic polyhedra in ℂ2 with a noncompact automorphism group

Let P be a bounded analytic polyhedron in ℂ² whose boundary is smooth except for normal crossing singularities. We show that P is a holomorphic quotient of the bidisc, if its automorphism group is noncompact.     

On proper holomorphic self-maps of generalized complex ellipsoids

The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in C ⁿ which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in C ⁿ such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing.     

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ?^k or an automorphism of a projective manifold.