Complex Euclidean Space Research Articles

Global Newlander–Nirenberg Theorem for Domains with C2 Boundary

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a C2 strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with C2 boundary D⊂Cn, we show the existence of global holomorphic coordinate systems defined on D‾ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small C2 perturbation of ∂D. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a C2 real hypersurface M⊂Cn, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

Conjugate phase retrieval on general Hilbert spaces

A conjugation on a Hilbert space means an antilinear bounded operator that the squares to the identity, which generalizes the traditional conjugation on complex Euclidean spaces. In this paper, with the help of normalized conjugation we introduce the notion of conjugate phase retrieval on general Hilbert spaces. We characterize frames that do conjugate phase retrieval; prove that every conjugate phase retrieval frame for consisting of all real vectors has the complement property in , and the converse is true if ; and also prove that a small perturbation of conjugate phase retrieval frame still gives a conjugate phase retrieval frame if , but it is false if .

A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

<abstract><p>In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.</p></abstract>

STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES

Previously, we have described the structure of the sets of Picard and Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros. It was proved that the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension not higher than p-1 of a p-dimensional complex Euclidean space. In addition, the sum of the dimensions of all these subspaces does not exceed p and any pairwise intersection of these subspaces contains only the zero vector. The set of Picard exceptional vectors has the same structure. For an integer curve of non-integer or zero order, the set of Borel exceptional vectors together with the zero vector is a single subspace of dimension no higher than p-1, which is not the case for the set of Picard exceptional vectors of an integer curve of non-integer order. The sufficiency of these results is confirmed to some extent by the corresponding examples. In this paper, we show that the above results on the structure of the set of Borel and Picard exceptional vectors for integer curves can be transferred to the case of analytic curves in the circle. We have also constructed an analytic curve in the circle, which confirms, to some extent, the sufficiency of the results. However, it was not possible to find out whether the corresponding result is transferred to the case of analytic curves of non-integer or zero order.

Approximation of holomorphic Legendrian curves with jet-interpolation

We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in C2n+1 lies on an injectively immersed isotropic surface with a prescribed complex structure. If the set has no accumulation points, the surface may be taken properly embedded. We also prove a Carleman-type theorem for holomorphic Legendrian curves with interpolation. Namely, a Legendrian curve, defined on a certain type of unbounded closed set in a given open Riemann surface R, may be approximated in the C0-topology by an entire Legendrian curve with prescribed finite-order Taylor polynomials at a closed discrete set of points in R. Under suitable conditions, the approximating map may be made into a proper embedding.

Holomorphic isometries into homogeneous bounded domains

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved by us earlier [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941] and by Xiaoliang Cheng and Yihong Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180].

ON SPECTRA OF SOME TENSORS OF SIX-DIMENSINAL KÄHLERIAN AND NEARLY-KÄHLERIAN SUBMANIFOLDS OF CAYLEY ALGEBRA

Six-dimensional Kählerian and nearly-Kählerian submanifolds of Cayley algebra are considered. Spectra of some classical tensors of such submanifolds of the octave algebra are computed. It is proved that a nearly-Kählerian six-dimensional submanifold of Cayley algebra is conharmonically flat if and only if it is holomorphically isometric to the complex Euclidean space with a canonical Kählerian structure.

Homogeneous Lagrangian foliations on complex space forms

We classify holomorphic isometric actions on complex space forms all of whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces.

Submanifolds of some Hartogs domain and the complex Euclidean space

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic.

The First Thirty Years of Andersén-Lempert Theory

In this paper we expose the impact of the fundamental discovery, made by Erik Anders\'en and L\'aszl\'o Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of complex Euclidean spaces of dimensions $n>1$. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present some focal points of these developments, with a view towards the future.

Algebraic properties of Hermitian sums of squares, II

We study real bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) in n n complex variables for which r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for r ( z , z ¯ ) r(z,\bar {z}) and the rank of r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of r r .

Codebook Training for Trellis-Based Hierarchical Grassmannian Classification

We consider classification of points on a complex-valued Grassmann manifold of m-dimensional subspaces within the n-dimensional complex Euclidean space. We introduce a trellis-based hierarchical classification network, which is based on an orthogonal product decomposition of the orthogonal basis representing the m-dimensional subspace. Exploiting the similarity of the proposed trellis classifier with a neural network, we propose stochastic gradient-based training techniques. We apply the proposed methods to two important applications in wireless communication, namely Grassmannian channel state information quantization in multiple-input multiple-output communications and non-coherent Grassmannian multi-resolution transmission.

The first eigenvalue for the p-Laplacian on Lagrangian submanifolds in complex space forms

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the [Formula: see text]-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the [Formula: see text]-Laplacian operator on closed orientate [Formula: see text]-dimensional Lagrangian submanifolds in a complex space form [Formula: see text] with constant holomorphic sectional curvature [Formula: see text]. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the [Formula: see text]-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for [Formula: see text] and [Formula: see text], respectively.

JASPER: a software for quantum chromodynamics Dyson-Schwinger equation numerical calculation

The JASPER program is the first part of the high-performance computing information system for estimate some elementary particle properties, developing at Petersburg Nuclear Physics Institute. The JASPER is an implementation of the Dyson-Schwinger equation numerical solution for simple dressed quark propagator calculation in rainbow approximation. The Dyson-Schwinger equation solution with the Marice-Tandy Ansatz is one of several phenomenological approaches to obtain quantitative results in quantum chromodynamics (QCD) within strong coupling regime. The JASPER program is programmed in the C++ language and uses the numerical algorithms from the GNU Scientific Library (GSL). The numerical results for dynamical quark mass in complex Euclidean space were obtained. This result will be employed to study the hadron spectrum with the Bethe-Salpeter equation approach.

A rank question for homogeneous polynomials

A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 > 4 n+1>4 that, while not sharp, is non-trivial.

НЕЛІНІЙНА КОНТАКТНА ДИНАМІКА ТА АНТИСИМЕТРІЯ КОРПУСКУЛЯРНО-ВИХОР-ХВИЛЬОВИХ ФОРМ ЕЛЕКТРОМАГНІТНОГО ТА ГРАВІТАЦІЙНОГО ПОЛІВ У ФОНОВОМУ СЕРЕДОВИЩІ КОМПЛЕКСНОГО ЕВКЛІДОВОГО ПРОСТОРУ. СПЕКТРИ ХІТОННОГО ВИПРОМІНЮВАННЯ

Based on the hydrodynamic-wave calibration of potentials in Maxwell’s equations and their analogues for the gravitational field, nonlinear equations with respect to the vector potentials of these fields in the background medium of a complex Euclidean space are obtained. The nonlinear contact dynamics of corpuscular-vortex-wave forms of fields and violation of antisymmetry, which leads to the formation of matter and generation of electromagnetic, gravitational, hydrodynamic , acoustic waves separately in real and imaginary half-spaces of complex Euclidean space, are considered. Analytical expressions for the spectra of heaton radiation in a complex Euclidean space are obtained. It is shown that these expressions describe, in particular, the spectrum of solar radiation, collider resonance spectra, the spectrum of microwave background radiation generated by the Oort Cloud, and other spectra in technical, space and geodynamic systems. The fundamental technical failures in the field of controlled thermonuclear fusion and the known catastrophes in nuclear energy and hydropower related to the disregard of corpuscular-wave dualism in macrosystems and the limitations of a purely real part of the complex Euclidean space are analyzed.

Real hypersurfaces of complex space forms satisfying Fischer–Marsden equation

Let M be a real hypersurface of a complex space form of constant curvature c. In this paper, we study the hypersurface M which admits a nontrivial solution to Fischer–Marsden equation, that is, the induce metric g of M satisfies $$ Hess _g(\nu )=(\varDelta _g \nu )g+\nu S_g$$ , where $$\nu $$ is a nontrivial function. We prove that there does not exist a complete Hopf real hypersurface in a non-flat complex space form satisfying Fischer–Marsden equation. Finally, we show that a complete real hypersurface with $$A\xi =\alpha \xi $$ , $$\alpha \ne 0$$ , of a complex Euclidean space $${\mathbb {C}}^n$$ satisfying Fischer–Marsden equation is locally congruent to a sphere or $${\mathbb {S}}^1 \times {\mathbb {R}}^{2n-2}$$ .

Compactness of the [formula omitted]-Neumann problem on domains with bounded intrinsic geometry

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the ∂¯-Neumann operator on (0,q)-forms is equivalent to the boundary not containing any q-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.