Weierstrass Formula Research Articles

$$\mathbb {C}P^{2S}$$ Sigma Models Described Through Hypergeometric Orthogonal Polynomials

The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb{C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any such projector solutions of the $\mathbb{C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrised in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb{C}P^{2S}$ models defined using the generalised Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the $\mathfrak{su}(2s+1)$ algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the $\mathfrak{su}(2)$ spin-s representation and the $\mathbb{C}P^{2S}$ model is explored in detail.

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

A generalized Weierstrass representation of Lorentzian surfaces in ℝ2,2 and applications

We give a generalized Weierstrass formula for a Lorentz surface conformally immersed in the four-dimensional space [Formula: see text] using spinors and Lorentz numbers. We also study the immersions of a Lorentzian surface in the Anti-de Sitter space (a pseudo-sphere in [Formula: see text]): we give a new spinor representation formula and deduce the conformal description of a flat Lorentzian surface in that space.

Soliton surfaces associated with sigma models: differential and algebraic aspects

In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the sigma model with finite action, defined in the Riemann sphere, are themselves solutions of the Euler–Lagrange equations for sigma models. On the other hand, we show that the Euler–Lagrange equations for surfaces immersed in the Lie algebra with conformal coordinates, that are extremals of the area functional, subject to a fixed polynomial identity, are exactly the Euler–Lagrange equations for sigma models. In addition to these differential constraints, the algebraic constraints, in the form of eigenvalues of the immersion functions, are systematically treated. The spectrum of the immersion functions, for different dimensions of the model, as well as its symmetry properties and its transformation under the action of the ladder operators are discussed. Another approach to the dynamics is given, i.e. description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model.

Analysis of sigma models via projective structures

This paper represents a study of projector solutions to the Euclidean sigma model in two dimensions and their associated surfaces immersed in the su(N) Lie algebra. Any solution for the sigma model defined on the extended complex plane with finite action can be written as a raising operator acting on a holomorphic one. Here the proof is formulated in terms rank-1 projectors so it is explicitly gauge invariant. We apply these results to the analysis of surfaces associated with the models defined using the generalized Weierstrass formula for immersion. We show that the surfaces are conformally parametrized by the Lagrangian density, with finite area equal to the action of the model, and express several other geometrical characteristics of the surface in terms of the physical quantities of the model. Finally, we provide necessary and sufficient conditions that a surface be related to a sigma model.

Weierstrass Formula for Minimal Surface in the Special Three-Dimensional Kenmotsu Manifold K with n-Parallel Ricci Tensor

Weierstrass Formula for Minimal Surface in the Special Three-Dimensional Kenmotsu Manifold K with n-Parallel Ricci Tensor

SURFACES OBTAINED FROM ℂPN-1 SIGMA MODELS

In this paper, the Weierstrass technique for harmonic maps S2 → ℂPN-1 is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the ℂPN-1 model equations are defined on the sphere S2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su (N) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N - 1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su (3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the ℂP2 model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semiinfinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in ℝ8 and ℝ3, respectively.

Conformally parametrized surfaces associated with sigma models

Two-dimensional parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the sigma model. The Lie-point symmetries of the model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in . The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.

Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator Δ 2 n , n ∈ N , acting on L 2 [ 0 , π ] with appropriate boundary conditions, generates an exponentially bounded K n -convoluted cosine function, and consequently, an exponentially bounded analytic K n + 1 -convoluted semigroup of angle π 2 , for suitable exponentially bounded kernels K n and K n + 1 .

Surfaces immersed in Lie algebras obtained from the sigma models

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.

Generalized Weierstrass representation of surfaces in [formula omitted

In the present paper, we describe the conformal immersion of the surface into R 4 by means of a linear system. Furthermore we prove that every regular conformal immersion of a surface into R 4 is locally determined by the generalized Weierstrass formulae. We also give the representation of the surface with parallel mean curvature vector by solutions with the parameter of a linear system which is determined by the sinh–Laplace equation.

Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field

A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.

On rigid string instantons in four dimensions

Generalized Weierstrass formulae for surfaces in four-dimensional space R 4 are used to study (anti)self-dual rigid string configurations. It is shown that such configurations are given by superminimal immersions into R 4 . Explicit formulae for generic (anti)instantons are presented. Particular classes of surfaces are also analyzed.

Harmonic two-spheres in compact symmetric spaces, revisited

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.

Integrated Semigroups and Related Partial Differential Equations

We investigate the relationship between abstract linear evolution equations of heat, wave, and Schrödinger types in terms of well-posedness in Banach spaces. More precisely, we study our operators as generators of integrated semigroups and integrated cosine functions. As applications, we consider in a Banach space context an abstract Laplacian which generalizes the ordinary Laplacian Δ in the L p ( R N )-spaces. We obtain optimal results when compared to the classical situation where the generation results were obtained by 17 using completely different methods. Our main tools are the boundary value theorem for holomorphic semigroups and the abstract Weierstrass formula. Related partial differential equations have been considered previously by Bragg and Dettman.

Weierstrass formula and zero-finding methods

Classical Weierstrass' formula [29] has been often the subject of investigation of many authors. In this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions. We are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros. Conditions for the safe convergence of Weierstrass' method, depending only on initial approximations, are given. In particular, we study polynomials with interval coefficients. Using an interval version of Weierstrass' method enclosures in the form of disks for the complex-valued set containing all zeros of a polynomial with varying coefficients are obtained. We also present Weierstrass-like algorithm for approximating, simultaneously, all zeros of a class of analytic functions in a given closed region. To demonstrate the proposed algorithms, three numerical examples are included.

A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in R3 into the hyperbolic 3-space.

A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in R3 into the hyperbolic 3-space.

Wellposedness of abstract Cauchy problems for second order differential equations

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A k)] toX, associated with an exponentially wellposed ACP onD(A k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H k) onX.

Weierstrass formula for surfaces of prescribed mean curvature

Weierstrass formula for surfaces of prescribed mean curvature