Browsing by Browse by FOR 2008 "010111 Real and Complex Functions (incl Several Variables)"

The aim of this article is to show how the individual harmonic components of the torsion of the canonical Cartan connection of embedded hyperbolic and elliptic CRmanifolds at a given point can be read off from the third order terms of the defining equation given in normal form. The general theory ensures that the vanishing of each of these one-dimensional components implies striking geometric consequences and we link each of them to an easily computable coefficient in the normal form. This allows to correct a mistake in [SS00] where it was claimed that four torsion components out of six vanish automatically for embedded CR-manifolds. The failure in that article appears already in Lemma 1.1 where the second order osculation was not dealt with carefully enough. At the same time, the rest of [SS00] is essentially worked out for abstract CR–structures and so the validity of the procedures and results has not been effected in general. In what follows, we use the terminology and notation of [SS00] without further comments.

In this paper we give the complete explicit description of the holomorphicautomorphisms of any nondegenerate CR-quadric Q of arbitrary CRdimensionand codimension and of Siegel domains of second kind with not necessarilyLevi-nondegenerate Silov-boundary.We introduce a family of k-dimensional chains (k = codim Q), the analoguesof one-dimensional Chern-Moser chains for hyperquadrics.We also analyse some different types of rigid quadrics.

Let M be a three–dimensional contact manifold, and ψ : D \{0} → M x ℝ a finite–energy pseudoholomorphic map from the punctured disc in ℂ that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that ψ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into Ep,q x R, where Ep,q denotes a rational ellipsoid (contact structure induced by the standard complex structure on ℂ²), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.

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For CR-manifolds in ℂ⁴ with the Levi form at the origin of parabolic type we construct an analogue of the Chern-Moser normal form for Levi non-degenerate hypersurfaces. The group of transformations which map a given CR manifold of parabolic type into a normal form is shown to be isomorphic with the isotropy group of the osculating parabolic quadric at the origin.

We give a complete classification of Levi-degenerate hypersurfaces of finite type in ℂ² with two-dimensional symmetry groups. Our analysis is based on the classification of two-dimensional Lie algebras and an explicit description of isotropy groups for such hypersurfaces, which follows from the construction of Chern–Moser type normal forms at points of finite type, developed in [M. Kolář, 'Normal forms for hypersurfaces of finite type in ℂ²', Math. Res. Lett. 12 (2005), pp. 897–910].

We explore the relationship between contact forms on S³ defined by Finsler metrics on S² and the theory developed by H. Hofer, K.Wysocki and E. Zehnder (Hofer et al. Ann. Math. 148, 197–289, 1998; Ann. Math. 157, 125–255, 2003). We show that a Finsler metric on S² with curvature K ≥ 1 and with all geodesic loops of length >π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J -holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on S² with K = 1, thus complementing the results obtained in Harris and Wysocki (Trans. Am. Math. Soc., to appear).

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not.

In the present paper we suggest an explicit construction of a Cartan connection for an elliptic or hyperbolic CR manifold M of dimension six and codimension two, i.e. a pair, consisting of a principal bundle over M and of a Cartan connection form over P, satisfying the following property: the (local) CR transformations are in one to one correspondence with the (local) automorphisms for which. For any, this construction determines an explicit monomorphism of the stability subalgebra Lie (Aut(M)x) into the Lie algebra of the structure group H of P.

In this paper, we present a complete list of rigid spherical CR hypersurfaces in C². The construction is based on a renormalization of Stanton's family of rigid spheres.

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n² are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n²)-dimensional submanifolds in C n+n² for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.

It is well known from undergraduate complex analysis that holomorphic functions of one complex variable are fully determined by their values at the boundary of a complex domain via the Cauchy integral formula. This is the first instance in which students encounter the general principle of complex analysis in one and several variables that the study of holomorphic objects often reduces to the study of their boundary values. The boundaries of complex domains, having odd topological dimension, cannot be complex objects. This motivated the study of the geometry of real hypersurfaces in complex space. In particular, since all established facts about a particular hypersurface carry over to its image via a biholomorphic mapping in the ambient space, it is important to decide which hypersurfaces are equivalent with respect to such mappings - that is, to solve an equivalence problem for real hypersurfaces in a complex space.

The main result of the paper is the following generalization of Forelli's theorem (Math. Scand. 41:358–364, 1977): Suppose 'F' is a holomorphic vector field with singular point at 'p', such that 'F' is linearizable at 'p' and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at 'p' and is holomorphic along the complex integral curves of 'F' is holomorphic in a neighborhood of 'p'. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.

The purpose of this paper is to present a generalization of Forelli's theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka (Kompleks. Anal. i Prilozh, 232-240, 2006) of 2005.

We use the method of model surfaces to study real four-dimensional submanifolds of ℂ³. We prove that the dimension of the holomorphic symmetry group of any germ of an analytic four-dimensional manifold does not exceed 5 if this dimension is finite. (There are only two exceptional cases of infinite dimension.) The envelope of holomorphy of the model surface is calculated. We construct a normal form for arbitrary germs and use it to give a holomorphic classification of completely non-degenerate germs. It is shown that the existence of a completely non-degenerate CR-structure imposes strong restrictions on the topological structure of the manifold. In particular, the four-sphere S⁴ admits no completely non-degenerate embedding into a three-dimensional complex manifold.

Let X be a complex n-dimensional reduced analytic space with isolated singular point x₀,and with a strongly plurisubharmonic function p : X --> [0;∞) such that p(x₀) = 0.A smooth Kähler form on X {x₀} is then defined by i p.The associated metric is assumed to have Lⁿ_loc-curvature, toadmit the Sobolev inequality and to have suitable volume growth near x₀.Let E --> X {x₀} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0,1)-form with coefficients in E.The main result of this article states that if ξ and the curvature of E are both Lⁿ_loc,then the equation ∂u = ξ has a smooth solution on a punctured neighbourhood of x₀.Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

The third Japanese-Australian Workshop on Real and Complex Singularities (JARCS SYDNEY 2009) was held at the University of Sydney, Australia, during the period 15-18 September 2009. There were 33 participants, mostly from Japan and Australia. The workshop covered a variety of topics in singularity theory and brought together experts, early career researchers, and doctoral students from Australia, France and Japan. This volume contains research papers in real and complex singularities, algebraic geometry and three introductory Lectures on Ominimal structures. It is our hope that this volume reflects the lively research atmosphere of this conference.

In 1974 Chern and Moser [4] constructed normal forms for real-analytic hypersurfaces with non-degenerate Levi-form in ℂⁿ⁺¹. For a real-analytic hypersurface M in ℂ² this means that there are local coordinates z, w centered in 0 ϵ M such that the equation of M takes the form...

The existence of non-linearizable isotropic CR-automorphisms of Levi non-degenerate hypersurfaces in complex space is characteristic for quadrics. In this paper we discover elliptic CR-manifolds of CR-dimension and codimension two that are not related to the quadric and that have non-linearizable isotropic automorphisms. This is a new and unexpected phenomenon. A complete description of such manifolds is given.

We give a complete description of normal forms for real hypersurfaces of finite type in C² with respect to their holomorphic symmetry algebras. The normal forms include refined versions of the constructions by Chern-Moser [6], Stanton [20], Kolář [14]. We use the method of simultaneous normalisation of the equations and symmetries that goes back to Lie and Cartan. Our approach leads to a unique canonical equation of the hypersurface for every type of its symmetry algebra. Moreover, even in the Levi-degenerate case, our construction implies convergence of the transformation to the normal form if the dimension of the symmetry algebra is at least two. We illustrate our results by explicitly normalising Cartan's homogeneous hypersurfaces and their automorphisms.

It is known that the CR geometries of Levi non-degenerate hyper-surfaces Cⁿ and of the elliptic or hyperbolic CR sub-manifolds of co-dimension two in C⁴ share many common features. In this paper, a special class of normalized coordinates is introduced for any CR manifold M which is one of the above three kinds and it is shown that the explicit expression in these coordinates of an isotropy automorphism... is equal to the expression of a corresponding element of the automorphism group of the homogeneous model. As an application of this property, an extension theorem for CR maps is obtained.

This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when Ω ⊂ ℂⁿ corresponds to a sub-level set of a smooth, real-valued function ψ, such that the form w = i∂∂ψ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.

This article is the revised and expanded version of a conferencepaper presented at the 9th International Conference on DifferentialGeometry and its Applications, Prague, Czech Republic, August30th – September 3rd, 2004. The following is a summary of recent results of the author in collaboration with K. Wysocki, concerning topological classification of 'J'–holomorphic curves, asymptotic to a given periodic orbit within a closed three–dimensional contact manifold.

Let X be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let η be a smooth form of type (1.1) defined on X\{0}. The main result of this note is a criterion for solubility of the equation ∂∂u = η. This implies a criterion for triviality of a Hermitian– holomorphic line bundle (L,h) --> X\{0} in a neighbourhood of the origin.

Let [this equation] be a ∂-closed form on ℂⁿ x ℂ. If it is assumed that ψ has compact (specifically non-empty) support, then ψn+1(w,z) ≠ 0, for if the last component vanishes identically then this, together with the closedness of ψ, implies that [this second equation] vanishes identically, 1 ≤ k ≤ n. Hence ψ is the pullback of a form defined on ℂⁿ, which contradicts the compactness of support.

Variational methods have long been regarded as the mathematical foundation of both classical and quantum mechanics, and continue to supply much of the impetus of modern symplectic topology and geometry. Their application in Hermitian geometry is a more recent development, though of comparable importance. The following partial survey will set out to expose their role specifically on the theory of Hermitian-Einstein vector bundles, and in those aspects of conformal field theory which involve deformations of complex structure.

We show that Stanton's list of rigid spherical hypersurfaces in [5] is not complete and determine the parameters that uniquely correspond to all such hypersurfaces.

In this paper we study the infinitesimal automorphisms of a multi-contact structure given by 2n vector fields on ℝ²ⁿ+¹. We prove that the Lie algebra of infinitesimal automorphisms has finite dimension (rigidity) and discuss a homogeneous model which has the largest algebra of infinitesimal automorphisms.

The fourth Japanese-Australian Workshop on Real and Complex Singularities (JARCS4 KOBE 2011) was held at the Kobe Satellite of Hyogo University of Teacher Education during the period 22-25 November, 2011. There were 31 participants from Australia and Japan. The Australian and Japanese Singularity groups have built up strong research relationship in the past three decades. For instance, the blow-analytic theory introduced by Tzee-Char Kuo in Australia has been intensively developed in Japan. In addition, a lot of joint research works of both countries have been established in several topics related to real and complex singularities. The present volume mainly consists of the texts of the invited talks of the workshop. Some of them are joint works of Australians and Japanese. This volume contains original articles on real and complex singularities, topology of differentiable maps, openings of differentiable map-germs, the relationship between free divisors and holonomic systems, effective computation method of invariants of singularities, the application of singularity theory to differential geometry, the deformation theory of CR structures and differential equation with singular points. In these articles some new notions important for characterizations of singularities are introduced, and several new results are presented. New approaches to classical topics and new computation methods of singularities are also presented.

We present a survey on the effect of triviality of nil-isotropic automorphisms of real-analytic non-quadratic Levi non-degenerate CR manifolds. As a new result we prove that any isotropic automorphism of an elliptic CR-manifold in ℂ⁴ is uniquely determined by its first order derivatives at the origin and that the dimension of the corresponding Lie group does not exceed 6. Moreover, it is shown that apart from a very special family of exceptional manifolds any isotropic automorphism is determined by the restriction of its differential at the origin to the complex tangent space.

We study real analytic CR manifolds of CR dimension 1 and codimension 2 in the three-dimensional complex space. We prove that the germ of a holomorphic mapping between 'nonspherical' manifolds can be extended along any path (this is an analog of Vitushkin's germ theorem). For a cubic model surface ('sphere'), we prove an analog of the Poincaré theorem on the mappings of spheres into ℂ². We construct an example of a compact 'spherical' submanifold in a compact complex 3-space such that the germ of a mapping of the 'sphere' into this submanifold cannot be extended to a certain point of the 'sphere'.