Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl _n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \mathbbRⁿ{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl _n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl _n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.     

Annihilators for Harmonic Differential Forms Via Clifford Analysis

In this paper we describe completely the annihilators of harmonic differential forms into the Clifford analysis approach.     

Duality for Harmonic Differential Forms Via Clifford Analysis

The space HF _k (Ω) of harmonic multi-vector fields in a domain as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HF _k (E) being a compact set. It is proved that HF _k (E)^* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity. Research of the third author was supported by the FWO Research Network WO. 003. 01N, research of the fourth author was supported by the FWO “Krediet aan Navorsers: 1.5.106.02”     

Asymptotically Euclidean Ends of Ricci Flat Manifolds,and Conformal Inversions

We prove that a Ricci flat end of a Riemannian manifold is asymptotically Euclidean if it is obtained from a smooth metric by a conformal inversion. A number of consequences are discussed.     

On Harmonic and C-Harmonic 1-Differentiable Forms on Sasakian Manifolds

In this paper, we firstly extend some classical operators on Sasakian manifolds acting to 1-differentiable forms on Sasakian manifolds. Next in a similar manner with the study of C-harmonic forms, we define and extend such a study for the case of 1-differentiable forms on Sasakian manifolds.     

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms ${L^\ell_k}$ which may be considered as the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as factored polynomials in second-order differential operators. In the case that the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the ${L^\ell_k}$ in terms of the null spaces of mutually commuting second-order factors.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds

By using the conformal method, solutions of the Einstein-scalar field gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension (> 2) as well as low regularity initial data.     

Hyperbolic Manifolds, Harmonic Forms, and Seiberg–Witten Invariants

New estimates are derived concerning the behavior of self-dual harmonic 2-forms on a compact Riemannian 4-manifold with nontrivial Seiberg–Witten invariants. Applications include a vanishing theorem for certain Seiberg–Witten invariants on compact 4-manifolds of constant negative sectional curvature.     

Radiation Fields on Asymptotically Euclidean Manifolds

Abstract

F.G. Friedlander introduced the notion of radiation fields for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation fields.     

Operator Exponentials for the Clifford Fourier Transform on Multivector Fields in Detail

In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit \({i \in \mathbb{C}}\) is replaced by a multivector square root of ?1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator.     

Conformal Deformation on Manifolds With Boundary

We consider natural conformal invariants arising from the Gauss–Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them.     

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).     

Conformal Vector Fields on Finsler Warped Product Manifolds

Acta Mathematica Sinica, English Series - In this paper, we study the conformal vector fields on Finsler warped product manifolds. We obtain a system of equivalent equations that the conformal...     

Boundary Behavior of Harmonic Functions on Manifolds

In this paper, we mainly set up a kind of representation theorem of harmonic functions on manifolds with Ricci curvature bounded below and study non-tangential limits of harmonic functions.     

调和函数和完备流形的结构

主要通过讨论调和函数来研究完备流形的几何性质,并推广了[1,9]中的结果．     

The Conformal Willmore Functional: A Perturbative Approach

The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds (M,g)) is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds (?³,g _? )—where g _? is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds (M,g) of dimension three.     

Conformal Metrics on the Unit Ball in Euclidean Space

We study densities on the unit ball in euclidean space whichsatisfy a Harnack type inequality and a volume growth conditionfor the measure associated with . For these densities a geometrictheory can be developed which captures many features of thetheory of quasiconformal mappings. For example, we prove generalizationsof the Gehring-Hayman theorem, the radial limit theorem andfind analogues of compression and expansion phenomena on theboundary. 1991 Mathematics Subject Classification: 30C65.     

Non-existence for Harmonic Maps on Complete Noncompact Manifolds

In this paper, we prove a nonexistence theorem on harmonic maps. This generalizes the well-known Liouville-type theorem on harmonic maps due to S.Y. Cheng and H.I Choi.