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1.
We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based
on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville
property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover,
the groupoid action on the Poisson boundary of any invariant operator is always amenable. This approach subsumes as particular
cases numerous earlier results on amenability for groups, actions, equivalence relations and foliations. For instance, we
establish in a unified way topological amenability of the boundary action for isometry groups of Gromov hyperbolic spaces,
Riemannian symmetric spaces and affine buildings.
Dedicated to Hillel Furstenberg 相似文献
2.
Qiao Yuying Swanhild Bernstein Sirkka-Liisa John Ryan 《Journal d'Analyse Mathématique》2006,98(1):43-64
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.
Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province
(105129)
Research supported by Academy of Finland 相似文献
3.
The aim of this paper is to give a geometric approach to Tits' amalgam method to construct buildings and to initiate a study of hyperbolic buildings, i.e. whose types are reflexion systems of the real hyperbolic space. We construct lots of examples and study their cohomology at infinity. We construct CAT(–1) polyhedral complexes having big discrete parabolic groups of isometries. 相似文献
4.
Sirkka-Liisa Eriksson Marko Kotilainen Visa Latvala 《Advances in Applied Clifford Algebras》2007,17(3):425-436
Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish
the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular,
we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among
else Bloch spaces, Bergman-spaces, Besov-spaces, and Qp-spaces.
The second author was supported by the Finnish Cultural Foundation. 相似文献
5.
In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [13]. Our focus here is on the theory of Hardy‐Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman‐Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
6.
In this paper we obtain a necessary and sufficient condition for the harmonicity of the inverse of a harmonic diffeomorphism. As an application, we give explicit representations of reversible logharmonic diffeomorphisms. As another application, we connect the harmonicity of the inverse of a hyperbolic harmonic quasiconformal diffeomorphism with the Schoen conjecture. 相似文献
7.
In this paper, we introduce the p-adic Moufang condition for hyperbolic buildings of rank 3. It is the most obvious and simplest generalization of the p-adic Moufang condition for affine buildings, introduced in Part III of this sequence of papers. We show that p is very restricted, which confirms (but does not prove) the conjecture that no p-adic analogue is possible for the construction of Moufang (hyperbolic) buildings by Ronan and Tits. 相似文献
8.
Martin Werner Licht 《Foundations of Computational Mathematics》2017,17(4):1085-1122
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincaré–Friedrichs-type inequalities will be studied in a subsequent contribution. 相似文献
9.
Bertrand Rémy 《Geometriae Dedicata》2002,90(1):29-44
We first remark that Kac–Moody groups enable us to produce hyperbolic buildings – automatically endowed with nonuniform lattices. The main result then deals with groups whose buildings are trees or two-dimensional hyperbolic. It is a factorization theorem for abstract isomorphisms. It shows the existence of nonisomorphic Kac–Moody groups with the same associated isomorphism class of buildings. 相似文献
10.
Grafting is a method of obtaining new projective structures from a hyperbolic structure, basically by gluing a flat cylinder into a surface along a closed geodesic in the hyperbolic structure, or by limits of that procedure. This induces a map of Teichmüller space to itself. We prove that this map is a homeomorphism by analyzing harmonic maps between pairs of grafted surfaces. As a corollary we obtain bending coordinates for the Bers embedding of Teichmüller space.
11.
Deane Yang 《Transactions of the American Mathematical Society》2000,352(3):1021-1038
This paper presents some existence and uniqueness theorems for harmonic maps between complete noncompact Riemannian manifolds. In particular, we obtain as a corollary a recent result of Hardt-Wolf on the existence of harmonic quasi-isometries of the hyperbolic plane.
12.
In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building forces to be one of the examples arising from PSL3(K) for a locally finite local skewfield K. Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist. 相似文献
13.
Alina Vdovina 《Mathematische Zeitschrift》2002,241(3):471-478
We give an elementary construction of polyhedra whose links are connected bipartite graphs. In particular, we construct polyhedra
whose links are generalized m-gons. The polyhedra of this type are interesting because of their universal coverings, which are two-dimensional hyperbolic
buildings.
Received: 10 March 2001; final form: 26 November 2001/ Published online: 29 April 2002 相似文献
14.
15.
We exhibit sharp inequalities comparing the hyperbolic distance (and hyperbolic metric) to distances (and associated metrics)
defined via positve harmonic functions as well as bounded harmonic functions. In the simply connected case, all four inequalities
are identities. For the non-simply connected case, we determine precisely when equality can hold: for a pair of points in
a distance inequality, or a single point in a metric inequality.
Part of this work was completed during a research visit to the University of Tennessee that was supported in part by a Travel
Grant for Research from the C.P. Taft Foundation. The second author would like to thank the Department of Mathematics at the
University of Tennessee for the kind hospitality accorded him during his visit. 相似文献
16.
An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa (Kyushu J Math 52(1):167?C185, 1998) and Jiang (Chin Ann Math Ser. 8A:376?C383, 1987) states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper, Caddeo et?al. (Isr J Math 130:109?C123, 2002) showed that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic. 相似文献
17.
In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates. 相似文献
18.
Jiaping Wang 《Journal of Geometric Analysis》1998,8(3):485-514
We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between
two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application,
we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary
maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing
energy density. 相似文献
19.
We show that a noncompact, complete, simply connected harmonic manifold (M d, g) with volume densityθ m(r)=sinhd-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M 2d, g) with volume densityθ m(r)=sinh2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented. 相似文献
20.
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric); the harmonic oscillator; and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces. 相似文献