Even-dimensional Euclidean Spaces Research Articles

Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

On even-dimensional Euclidean space for integer powers of the positive Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on hyperspheres in arbitrary even dimensions.

Comparing volumes by concurrent cross-sections of complex lines: a Busemann–Petty type problem

We consider the problem of comparing the volumes of two star bodies in an even-dimensional Euclidean space $${\mathbb {R}}^{2n} = {\mathbb {C}}^n$$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann–Petty type theorem holds. Quaternionic and octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen’s inequality. Along the way we provide a criterion that detects which centered bodies are circular. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann–Petty type result with a minimum of required background (in the spirit of L. K. Hua’s book on the classical domains) and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.

A sharp lower bound for a resonance-counting function in even dimensions

This paper proves sharp lower bounds on a resonance-counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of -Δ on ℝ d , for example, for the Laplacian for certain metric perturbations on ℝ d . The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions.

Partially massless higher-spin theory II: one-loop effective actions

We continue our study of a generalization of the D-dimensional linearized Vasiliev higher-spin equations to include a tower of partially massless (PM) fields. We compute one-loop effective actions by evaluating zeta functions for both the “minimal” and “non-minimal” parity-even versions of the theory. Specifically, we compute the log-divergent part of the effective action in odd-dimensional Euclidean AdS spaces for D = 7 through 19 (dual to the a-type conformal anomaly of the dual boundary theory), and the finite part of the effective action in even-dimensional Euclidean AdS spaces for D = 4 through 8 (dual to the free energy on a sphere of the dual boundary theory). We pay special attention to the case D = 4, where module mixings occur in the dual field theory and subtlety arises in the one-loop computation. The results provide evidence that the theory is UV complete and one-loop exact, and we conjecture and provide evidence for a map between the inverse Newton’s constant of the partially massless higher-spin theory and the number of colors in the dual CFT.

Half Dirichlet problem for the Hölder continuous matrix functions in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed in the framework of circulant (2 × 2) matrix functions. Under this setting, we will present the half Dirichlet problem with boundary spaces of Hölder continuous circulant (2 × 2) matrix functions on the sphere of even-dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solutions to the Dirichlet problem on the unit ball for Hölder continuous circulant (2 × 2) matrix functions as the boundaries and the solutions to the classical Dirichlet problem as the special case, and derive a decomposition of the Poisson kernel for matrix Laplace operator.

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

In even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution for the polyharmonic equation has logarithmic behavior. We give two approaches for developing an azimuthal Fourier expan- sion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials which we call logarithmic polynomials. The second approach depends on the computation of parameter derivatives of Fourier series expressions for a power-law fundamental solution of the polyharmonic equation. We conclude by comparing the two approaches and giving the azimuthal Fourier series for a logarithmic fundamental solution of the polyharmonic equation in rotationally-invariant coordinate systems. Solutions of the polyharmonic equation (powers of the Laplacian operator) are ubiquitous in many areas of computational, pure, applied mathematics, physics and en- gineering. We concern ourselves, in this paper, with a fundamentalsolution of the poly- harmonic equation (Laplace, biharmonic, etc.), which by convolution yields a solution to the inhomogeneous polyharmonic equation. Solutions to inhomogeneous polyhar- monic equations are useful in many physical applications including those areas related to Poisson's equation such as Newtonian gravity, electrostatics, magnetostatics, quan- tum direct and exchange interactions (cf. Section 1 in (3)), etc. Furthermore, applica- tions of higher-powers of the Laplacian include such varied areas as minimal surfaces (12), Continuum Mechanics (8), Mesh deformation (6), Elasticity (9), Stokes Flow (7), Geometric Design (20), Cubature formulae (17), mean value theorems (cf. Pizzetti's formula) (13), and Hartree-Fock calculations of nuclei (21). Closed-form expressions for the Fourier expansions of a logarithmic fundamental solution for the polyharmonic equation are extremely useful when solving inhomoge- neous polyharmonic problems in even-dimensionalEuclidean space, especially when a degreeof rotationalsymmetry is involved. A fundamentalsolution of the polyharmonic equation on d-dimensionalEuclidean space R d has two argumentsand therefore maps from a 2d-dimensional space to the reals. Solutions to the inhomogeneous polyhar- monic equation can be obtained by convolution of a fundamental solution with an in- tegrable source distribution. Eigenfunction decompositions of a fundamental solution

Instantons and Killing spinors

We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.

Graphs associated with vector spaces of even dimension: A link with differential geometry

We define a new association between graphs and orthonormal bases of even-dimensional Euclidean vector spaces endowed with an special isomorphism motivated by the recently introduced theory of submanifolds associated with graphs. We provide several interesting examples and we analyze the shape of such graphs by proving some general results.

Bifurcation effects in sublinear elliptic problems on compact Riemannian manifolds

We prove a bifurcation result for a perturbed sublinear elliptic problem defined on a compact Riemannian manifold without boundary. This result is then applied to a singular elliptic problem on even-dimensional Euclidean spaces.

Pointwise convergence for expansions in spherical monogenics

We offer a new approach to deal with the pointwise convergence of Fourier-Laplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter's theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson's theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson's theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

We study the ground state of the Pauli Hamiltonian with a magnetic field in \({\mathbb{R}^{2d}}\) , d > 1. We consider the case where a scalar potential W is present and the magnetic field B is given by \({B=2i\partial\bar{\partial} W}\) . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.

Logarithmic potentials, quasiconformal flows, and Q-curvature

By using quasiconformal flows, we establish that exponentials of logarithmic potentials of measures of small mass are comparable to Jacobians of quasiconformal homeomorphisms of Rn, n≥2. As an application, we obtain the fact that certain complete conformal deformations of an even-dimensional Euclidean space Rn with small total Paneitz or Q-curvature are bi-Lipschitz equivalent to standard Rn

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.

SPHERICAL SIMPLICES AND THEIR POLARS

This article is about spherical simplices in the unit sphere. One of the purposes is to give a relation of dihedral angles of a spherical simplex and its polar, and the other is to give two simple formulae of volumes of spherical simplices and their polars. We can calculate the volume of a spherical simplex and the sum of volumes of it and its polar from these formulae for the unit sphere in the odd- and even-dimensional Euclidean spaces, respectively.

The derivation of the regularized chiral Jacobian using the zeta function method

Using the zeta function method, a general formula for the regularized chiral Jacobian to theories including non-Hermitian Dirac operators 𝒟 defined in arbitrary even-dimensional Euclidean space is derived. The agreement of this formula with the results obtained in the differential geometric approach is also clarified.

Generic submanifolds of an even-dimensional Euclidean space

Generic submanifolds of an even-dimensional Euclidean space

Submanifolds of codimension $2$ in an even-dimensional Euclidean space

Submanifolds of codimension $2$ in an even-dimensional Euclidean space

On normal $(f,\,g,\,u,\,v,\, \lambda)$-structures on submanifolds of codimension $2$ in an even-dimensional Euclidean space

It is well known that a hypersurface of an almost Hermitian manifold admits an almost contact metric structure naturally induced on it. The study of hypersurfaces of a Euclidean space and of a Kahlerian manifold on which the induced almost contact metric structure satisfies certain conditions has been started by one of the present authors [4, 5]. On the other hand Blair [1, 2], Goldberg [3], Ludden [1, 2], Yamaguchi [8] and the present authors [6, 9] started the study of hypersurface of an almost contact manifold and of submanifolds of codimension 2 of an almost complex manifold. These submanifolds admit, under certain conditions, what we call (/, g, u, v, λ) structure. An even-dimensional sphere of codimension 2 of an even-dimensional Euclidean space is a typical example, of a manifold which admits this kind of structure. In a previous paper [9], we have studied the (/, g, u, v, -structure and given characterizations of even-dimensional sphere. In the present paper, we study submanifolds of codimension 2 in an evendimensional Euclidean space which admit a normal (/, g, u, v, Λ)-structure. In § 1, we consider submanifolds of codimension 2 of an even-dimensional Euclidean space regarded as a flat Kahlerian manifold. In the next section, we deal with (/, g, u, v, -structure induced on a submanifold of codimension 2 of an even-dimensional Euclidean space. In § 3, we find differential equations which /, g, u, v and λ satisfy. § 4 is devoted to the study of relations between the structure equations of the submanifold and the induced (/, g, u, v, ^-structure. In § 5 we prove a series of lemmas which are valid for normal (/, g, u, v, λ}structures and in § 6 we study properties of the mean curvature vector of the submanifold with normal (/, g, u, v, -structure. In the last § 7, we study hypersurfaces of an odd-dimensional Euclidean space and determine all the hypersurfaces admitting a normal (/, g, u, v, Λ)-structure. Our main theorem appears at the end of §7.

Contact hypersurfaces in certain Kaehlerian manifolds

Introduction. An odd dimensional differentiable manifold M is called an almost contact manifold if the structure group of its tangent bundle can be reduced to (7(n)xl, where U(n) means the real representation of the unitarygroup of n complex variables. Recently S. Sasaki [12] proved that the condition of almost contact manifold can be represented by a set of tensor fields φ, f, η which satisfy certain conditions. From this fact it is known that any orientable hypersurface of an almost complex manifold admits the structure of an almost contact manifold. The study of hypersurfaces of an almost complex manifold is perhaps one of the most fruitful aspects of the theory of an almost contact manifold. In fact, M. Kurita [8], Y. Tashiro and S. Tachibana [15] and the present author [9,10,11] proved many theorems on certain hypersurface of an almost complex manifold. If the induced almost contact metric of a hypersurface of almost Hermitian manifold is contact metric we call the hypersurface a contact hypersurface. In this paper, we mainly study the differential geometric properties of contact hypersurfaces of Kaehlerian manifold of constant holomorphic sectional curvature. In particular, as an even-dimensional Euclidean space E can be regarded as a Kaehlerian manifold with vanishing sectional curvature, the results obtained apply to contact hypersurface of a Euclidean space. In §1, we give first of all the definition of an almost contact metric manifold and, in §2, a short summary of those parts of the theory of hypersurfaces in an almost Hermitian manifold which are necessary for what follows. Moreover in §2 some general properties of contact hypersurfaces of a Kaehlerian manifold are derived and in §3 some examples of contact hypersurfaces of a Kaehlerian manifold are exhibited. After these preliminaries we consider in §4 hypersurfaces of Kaehlerian manifold of constant holomorphic sectional curvature and prove that there are at most three distinct principal curvatures of a contact hypersurface in

An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces

An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces