Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

Geophysical Inverse Theory and Regularization Problems

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.)

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces.

This paper investigates the numerical computation of a Cauchy problem for Laplace's equation which is a typical ill-posed problem. By using Green's formula, the problem is transformed to a moment problem. For numerical computation of the moment problem, an error estimation and several numerical examples for verification are presented. Necessary and sufficient conditions for the existence of the solution of the Cauchy problems for Laplace's equation in two-dimension are also given.

Hyperbolic space is a well-defined space with constant negative curvature. Recent research demonstrates its odds of capturing complex hierarchical structures with its exceptional high capacity and continuous tree-like properties. This paper bridges hyperbolic space's superiority to the power-law structure of documents by introducing a hyperbolic neural network architecture named Hyperbolic Hierarchical Attention Network (Hype-HAN). Hype-HAN defines three levels of embeddings (word/sentence/document) and two layers of hyperbolic attention mechanism (word-to-sentence/sentence-to-document) on Riemannian geometries of the Lorentz model, Klein model and Poincaré model. Situated on the evolving embedding spaces, we utilize both conventional GRUs (Gated Recurrent Units) and hyperbolic GRUs with Möbius operations. Hype-HAN is applied to large scale datasets. The empirical experiments show the effectiveness of our method.

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

Given a hyperbolic manifold M and an embedded tube of radius r about some geodesic, we determine an upper bound on the percentage of the volume of M occupied by the tube.

This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.

In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.

A metric tree is a metric space such that between any two points there is a unique topological segment (i.e., a subset homeomorphic to an interval of the real line), and such that this segment is a geodesic (i.e., the isometric image of an interval). The theory of group actions on metric trees and of the structure of these trees on the boundary at infinity has many different aspects, and it has been made clear that this theory captures a lot of the interesting phenomena of hyperbolic geometry, or, to be more precise, of the theory of groups acting by isometrics on the hyperbolic space I/~ and on its sphere at infinity. One of the particular features, in the case of metric trees, is that the ideas and methods which are used are often of a combinatorial nature, and, probably, simpler and more intuitive than those of classical hyperbolic geometry. In particular, in [2] and [3], we give a series of criteria for the ergodicity of the geodesic flow associated with groups acting properly and isometrically on metric trees, which are discrete analogs of theorems due to Hopf and Sullivan in the case of hyperbolic manifolds. The invariant measures on these flows come from conformal densities on the boundary. This theory of conformal densities on the boundary was developed, in the case of hyperbolic manifolds, by Patterson and Sullivan (see [5] and [6]), and later, in the case of metric trees, in [11 . The result that we prove here concerns a Fatou-type behavior of certain natural functions associated with conformal densities on the boundary of a metric tree. A somehow more special result of this kind, in the case of conformal densities on the boundary of the hyperbolic n-space, is used by Sullivan in [6], in his proof for the probabilistic criterion of the ergodicity of the geodesic flow on a hyperbolic manifold (see also [7, Theorem 2.14]). T h e p r o o f of the corresponding result for metric trees, which we give in [31, involves a different kind of argument, although the Fatou-type behavior is also valid in that case, and, indeed, the aim of the present paper is to prove this result in the case of metric trees. Let us begin by describing the background. Let X be a metric tree which is complete and locally compact. We denote the distance between two points in X by Ix YlA geodesic ray (resp. a geodesic segment) is a map f from [0, oo[ (resp. from a closed interval) into X such that I f(r t ) f(r2)l = Irt r~l for every r, and r2. The boundary OX is the space of the ends of X. Every geodesic ray r defines a point in OX, which is denoted by r(oo). Let us also recall that between any two points x E X and r /E OX, there is a unique image of a geodesic ray, which is denoted by Ix, r/[. There is a natural topology on X U OX, which can easily be defined with the use of the ray structure from any given point in X, and for this topology, the space X U OX is compact and X lies in i t as an open dense subset. Better than that, X tO OX is, in fact, equipped with a natural class of metrics, and the induced class of metrics on OX, called visual metrics, is most useful. Let us recall its definition: For every point z E X, the visual metric I I~ on OX is defined by

By a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume We recall that if is a Kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space Otherwise is geometrically in nite If happens to be a surface group then we say it is quasi Fuchsian if the limit set for the group action is a Jordan curve C and preserves the components of S n C The starting point for this work is the following theorem which is a combination of theorems due to Marden Thurston and Bonahon

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.

Voronoi diagrams belong to frequently used structures in computational geometry with application in many fields of science. The properties of Voronoi diagram are already studied in various metric spaces – Euclidean, Manhattan, Minkowski, Hausdorff, or Karlsruhe and also in the hyperbolic metric. In this paper, we focus on the Voronoi diagram and its dual in the Poincaré ball model of the three-dimensional hyperbolic space. We first present some basic tools from the Poincaré ball model needed to construct a Voronoi diagram and for a closer observation of its properties. We have determined certain conditions for the position of the generators controlling the behavior of the hyperbolic Voronoi diagram. In the last section, we demonstrate this effect on the dual graph of a hyperbolic Voronoi diagram, i. e., on hyperbolic Delaunay tessellations.KeywordsPoincaré ball modelhyperbolic spaceVoronoi diagramDelaunay tessellation