General Integral Formulas for k-hyper-mono-genic Functions

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC 1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

The aim of this article is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function x m is included. The leading idea is that the power function is the conjugate gradient of a harmonic function, defined with respect to the hyperbolic metric of the upper half space. In this paper we give a new approach to this hyperbolic function theory and survey some of its results.

We give sufficient conditions which ensure that harmonic extension u = P[f] to the upper half space {(x, y) | x ? Rn, y > 0} of a function f ? Lp(Rn) satisfies estimate ?u/?y (x, y) ? C?(y)/y for every x in E ? Rn, where ? is a majorant. The conditions are expressed in terms of behaviour of the Riesz transforms Rjf of f near points in E. We briefly investigate related questions for the cases of harmonic and hyperbolic harmonic functions in the unit ball.

We introduce a new notion called the extended hyperbolic metrics, as a hyperbolic metric (i.e. metric of constant curvature − 1) with certain kinds of singularities defined on a Riemann surface, and we give several fundamental properties of such metrics. Extended hyperbolic metrics are closely related to space-like surfaces of constant mean curvature one (i.e. CMC-1 surfaces) in de Sitter 3-space S 1 3. For example, the singular set of a given CMC-1 surface in S 1 3 is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S 1 3 (i.e. weakly complete constant mean curvature 1 surfaces in S 1 3 of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S 1 3-catenoids. Since there is a bijection between the moduli space of S 1 3-catenoids and the moduli space of co-orientable extended hyperbolic metrics with two regular singularities, a classification of such hyperbolic metrics is also given. (Co-orientability of extended hyperbolic metrics is defined in this paper.)

The main goal of this paper is to give the basic principles of hyperbolic function theory. First, we discuss the Green identity for . Second, we also deal with the integral representations for hyperbolic harmonic function in the Clifford algebra

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique, Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This conjecture was proved to be false by Farrell and Jones [6] in every dimension in which exotic spheres exist. They constructed examples of homeomorphisms f : M → N between closed negatively curved manifolds for which f is not homotopic to a diffeomorphism. These counterexamples were later obtained also in dimension six by Ontaneda [21] and later generalized by Farrell, Jones and Ontaneda to all dimensions > 5 [8]. In fact, in [21] and [8] examples are given for which f is not even homotopic to a PL homeomorphism. The fact that f is not homotopic to a PL homeomorphism has several interesting strong consequences that imply certain limitations of well known powerful analytic ethods in geometry [9], [10], [11], [12] (see [13] for a survey). In all the examples mentioned above, one of the manifolds is always a closed hyperbolic manifold. Hence, both manifolds M and N have the homotopy type of a closed hyperbolic manifold (hence the homotopy type of a closed locally symmetric space). We call these examples of hyperbolic homotopy type. In [2], Ardanza also gave counterexamples to the Lawson–Yau conjecture. In his examples, the manifolds M and N are not homotopy equivalent to a closed locally symmetric space; in particular, they are not homotopy

Let Cℓ n be the (universal) Clifford algebra generated by e1,…,e n The Cauchy-Riemann (or Dirac) operator in Cℓ n is defined by $$ D = \sum\nolimits_{{i = 0}}^{n} {{{e}_{i}}} \frac{\partial }{{\partial {{x}_{i}}}}. $$ The functions $$ f{\text{ }}:{\text{ }}\Omega \to C{{\ell }_{{n,}}}{\text{ }}\Omega {\text{ }} \subset {\text{ }}{{\mathbb{R}}^{{n + 1}}} $$ open, satisfying Df = 0, called left monogenic functions, are widely studied in Clifford analysis. The power function xm is not monogenic. The second author noticed that the power function is the solution of the modified Cauchy-Riemann system $$ xnDf + \left( {n - 1} \right)fn = 0 $$ which has strong connections to the hyperbolic metric. We study solutions of the equation $$ xnDf + \left( {n - 1} \right)Q_{{n - 1}}^{'}\left( f \right) = 0, $$ where Qf is the function given by the decomposition $$ f\left( x \right) = Pf\left( x \right) + Qf\left( x \right){{e}_{n}} $$ for $$ Pf\left( x \right),Qf\left( x \right) \notin C{{\ell }_{{n - 1}}}. $$ If $$ f = \sum\nolimits_{{i = 0}}^{n} {{{f}_{i}}} {{e}_{i}} $$ for some real functions f i , then f is the solution of the modified Cauchy-Riemann system stated above.

We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature.

Chapter 8 - Clifford algebra approaches to mechanical geometry theorem proving

Using the Clifford algebra C n , with generators $$ {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, $$ the Cauchy-Riemann operator in ℝ n+1 is defined as $$ \bar{\partial } = \frac{\partial }{{\partial xo}} + \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}{e_{i}}} , $$ where $$ x = {x_{0}} + \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , $$ is a paravector. We consider Fueter-type paravector functions f = u( x o, p) + I( x) v( x o, p), where $$ {p^{2}} = \sum\limits_{{i = 1}}^{n} {x_{i}^{2}} ,I\left( x \right): = \frac{1}{p} = \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , $$ and u, v are real-valued. The equation $$\bar{\partial }f = 0$$ splits into two parts. One of them depends only on xo,p. This leads to a system of partial differential equations which coincides with the system defining hypermonogenic functions. These functions arise for example as solutions of the Dirac equation in the upper half space ℝ + +1 endowed with the Poincaré metric.

Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝ n and let Har0(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx 0,x 1 ∈U/Γ such that the reflection law (RL)u(x 0)+Ku(x 1)=0, ∀u∈Har0(U, Γ) holds for some constantK. This is equivalent to the class Har0 (U, Γ) not separating the pointsx 0,x 1. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.

In this paper we present some results on the asymptotic growth behavior of periodic paravector valued functions that satisfy the Dirac‐Hodge equation on upper half‐space. We set up a generalization of the classical Valiron inequality for this class of functions and discuss some basic properties.

Inclusion functions for complex standard functions have been implemented in the CoStLy C++ class library. Among other functions, CoStLy includes procedures for the exponential function, the logarithm, root and power functions, trigonometric and hyperbolic functions, and their inverse functions.Numerical examples are presented to demonstrate the high level of accuracy in computation that has been achieved with our implementation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)