Complex Euclidean Space Research Articles

The Oka Principle for Sections of Stratified Fiber Bundles

A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.

Functions of orthogonal projectors involving the Moore–Penrose inverse

Several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors. The results were established in various settings, including finite and infinite dimensional vector spaces over either real or complex numbers, but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors. In the present paper, these results are unified and reestablished in a generalized form in a complex Euclidean vector space, with the generalization obtained by relaxing the nonsingularity assumption and use of the Moore–Penrose inverse instead of the ordinary inverse. Additionally, several new formulae of the type are provided.

Isotropic Lagrangian Submanifolds in Complex Euclidean Space and Complex Hyperbolic Space

The notion of isotropic submanifolds of an arbitrary Riemannian manifold was first introduced by B. O’Neill in [12]. In our paper, we give a complete classification of isotropic Lagrangian submanifolds in complex Euclidean space \({\mathbb{C}}^n\) and complex hyperbolic space \({\mathbb{CH}}^n\).

A construction of Lagrangian submanifolds in complex Euclidean spaces with legendre curves

In [1], B. Y. Chen provided a new method to construct Lagrangian surfaces in C2 by using Legendre curves in S3(1) $\subset$ C2. In this paper, we investigate the similar methods to construct some Lagrangian submanifolds in complex Euclidean spaces Cn (n ≥ 3).

Cyclic and ruled Lagrangian surfaces in Euclidean four space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-space. We also describe ruled Lagrangian surfaces. Finally we characterize those cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces.

EFFECTIVE ŁOJASIEWICZ INEQUALITY FOR ARITHMETICALLY DEFINED VARIETIES AND A GEOMETRIC APPLICATION TO BIHOMOGENEOUS POLYNOMIALS

We establish two versions of effective Łojasiewicz inequality for arithmetically defined affine varieties. As an application, we consider bihomogeneous polynomials on the complex Euclidean space which are positive along the affine cone of an arithmetically defined projective variety, and we obtain effective estimates on certain modifications needed to turn them into sums of squares of pointwise norms of homogeneous polynomials. The latter can be interpreted as an effective result on isometric embeddings for the associated indefinite Hermitian holomorphic line bundles.

COMPLEX SUBMANIFOLDS OF ALMOST COMPLEX EUCLIDEAN SPACES

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of (ℝ4, J), for some almost complex structure J if and only if it is an elliptic curve. Furthermore, we show that any (almost) complex 2n-torus can be holomorphically embedded in (ℝ4n, J) for a suitable almost complex structure J. This allows us to embed any compact Riemann surface in some almost complex Euclidean space and to show many explicit examples of almost complex structures in ℝ2n, which cannot be tamed by any symplectic form.

Maximally Equiangular Frames and Gauss Sums

In a finite-dimensional complex Euclidean space, a maximally equiangular frame is a tight frame which has a number of elements equal to the square of the dimension of the space, and in which the inner products of distinct elements are of constant magnitude. Though the general question of their existence remains open, many examples of maximally equiangular frames have been constructed as finite Gabor systems. These constructions involve number theory, specifically Schaar’s identity, which provides a reciprocity formula for quadratic Gauss sums. To be precise, Zauner used Schaar’s identity to compute the spectrum of a chirp-Fourier operator, the eigenvectors of which he conjectured to be well-suited for the construction of maximally equiangular Gabor frames. We provide two new characterizations of such frames, both of which further confirm the relevance of the theory of Gauss sums to this area of frame theory. We also show how the unique time-frequency properties of a particular cyclic chirp function may be exploited to provide a new, short and elementary proof of Schaar’s identity.

Super-conformal surfaces associated with null complex holomorphic curves

A correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in fourdimensional Euclidean space is defined by the quaternionic theory of surfaces. As an application, a transformation of super-conformal surfaces is defined.

CHEN IDEAL KAEHLER HYPERSURFACES

The concept of Chen ideal submanifolds is illustrated by characterizing the complex hypercylinders in the complex Euclidean spaces and the complex hyperquadrics in the complex projective space forms in terms of equalities involving their intrinsic and normal scalar curvatures and simplest “delta” curvatures.

Analysis and Geometry on Worm Domains

In this primarily expository article, we study the analysis of the Diederich-Fornæss worm domain in complex Euclidean space. We review its importance as a domain with nontrivial Nebenhülle, and as a counterexample to a number of basic questions in complex geometric analysis. Then we discuss its more recent significance in the theory of partial differential equations: the worm is the first smoothly bounded, pseudoconvex domain to exhibit global non-regularity for the $\overline{\partial}$ -Neumann problem. We take this opportunity to prove a few new facts. Next, we turn to specific properties of the Bergman kernel for the worm domain. An asymptotic expansion for this kernel is considered, and applications to function theory and analysis on the worm are provided.

Standardization of certain compact group actions and the automorphism group of the complex Euclidean space

For a connected compact Lie group of rank n, we study its effective action on an n-dimensional complex manifold as a holomorphic transformation group. We give a result on the standardization of such an action, called the generalized standardization theorem, by using the theory of Reinhardt domains and a result of Barrett, Bedford and Dadok. The generalized standardization theorem has various applications. As one of them, we apply it to the study of the structure of the holomorphic automorphism group Aut(C n ) of C n , and prove two theorems on subgroups of Aut(C n ). One is a fact on the conjugacy of compact subgroups of Aut(C n ) whose rank is n, and the other is about the nonexistence of a certain noncompact subgroup of Aut(C n ).

Warped product Kähler manifolds and Bochner–Kähler metrics

Using as an underlying manifold an alpha-Sasakian manifold, we introduce warped product Kähler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kähler manifold is of quasi-constant holomorphic sectional curvatures with a special distribution. Conversely, we prove that any Kähler manifold of quasi-constant holomorphic sectional curvatures with a special distribution locally has the structure of a warped product Kähler manifold whose base is an alpha-Sasakian space form. As an application, we describe explicitly all Bochner–Kähler metrics of quasi-constant holomorphic sectional curvatures. We find four families of complete metrics of this type. As a consequence, we obtain Bochner–Kähler metrics generated by a potential function of distance in complex Euclidean space and of time-like distance in the flat Kähler–Lorentz space.

CHEN IDEAL KAEHLER HYPERSURFACES

The concept of Chen ideal submanifolds is illustrated by characterizing the complex hypercylinders in the complex Euclidean spaces and the complex hyperquadrics in the complex projective space forms in terms of equalities involving their intrinsic and normal scalar curvatures and simplest “delta” curvatures.

Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits

Special Lagrangian cones in complex Euclidean spaces are obtained as cones over compact minimal Legendrian submanifolds in the odd dimenisonal standard hypersphere. The notion of the stability, the Legendrian stability and the rigidity of special Lagrangian cones were recently introduced and investigated by D. Joyce, M. Haskins etc. In this paper we determine explicitly the stability-index, the Legendrian-index, and the rigidity of special Lagrangian cones over compact irreducible symmeric spaces of type $A$ obtained as minimal Legendrian orbits and over a minimal Legendrian $\mathit{SU}(2)$-orbit. We obtain the examples of stable and rigid special Lagrangian cones in higher dimensions. Moreover we discuss a relationship of these properties with the Hamiltonian stability of minimal Lagrangian submanifolds in complex projective spaces.

Chern-Lashof absolute curvature of complex submanifolds and volumes of Grassmann images

We introduce an analog of the Chern-Lashof absolute curvature for complex submanifolds in complex Euclidean spaces. A relation between this curvature and the volume of the Grassmann image of the submanifold is established.

Fine structure for 3D second-order superintegrable systems: three-parameter potentials

A classical (or quantum) superintegrable system of second order is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion quadratic in the momenta, the maximum possible. For n = 3 on conformally flat spaces with nondegenerate, i.e., four-parameter potentials (the extreme case), we have worked out the structure and classified most of the possible spaces and potentials. Here, we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems, the three-parameter potential case. We show that for ‘true’ three-parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. This is a significant step towards the complete structure analysis of all types of second-order superintegrable systems.

REAL HYPERSURFACES IN COMPLEX SPACE FORMS WITH ε-PARALLEL RICCI TENSOR AND STRUCTURE JACOBI OPERATOR

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]). In this paper we investigate real hypersurfaces M in a nonflat complex space form Mn(c) under the condition that ∇ξS = 0 and ∇ξRξ = 0, where S and Rξ respectively denote the Ricci tensor and the structure Jacobi operator of M in Mn(c). 0. Introduction A Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). It is well known that complete and simply connected complex space forms are isometric to a complex projective space PnC, a complex Euclidean space C or a complex hyperbolic space HnC according as c > 0, c = 0 and c < 0. In this paper we consider a real hypersurface M in a complex space form Mn(c), c6=0. Then M has an almost contact metric structure (φ, ξ, η, g) induced from the complex structure J and the Kaehler metric of Mn(c). The structure vector field ξ is said to be principal if Aξ = αξ is satisfied, where A denotes the shape operator of M and α = η(Aξ). A real hypersurface is said to be a Hopf hypersurface if the structure vector field ξ of M is principal. In the study of real hypersurfaces in PnC, Takagi [12] classified all homogeneous real hypersurfaces and Cecil and Ryan [2] showed that they can be regarded as the tubes of constant radius over Kaehler submanifolds when the structure vector field ξ is principal. Such tubes can be divided into six kinds of type A1, A2, B, C, D and E. Received August 8, 2005; Revised May 30, 2006. 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

Nondegenerate superintegrable systems in n-dimensional Euclidean spaces

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained.

Exceptional and non-crystallographic root systems and the Kochen–Specker theorem

The Kochen–Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases E6 and E7 lead to new examples, while F4, E8 and H4 yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space).