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A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in R n , we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP. [ABSTRACT FROM AUTHOR]

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling Y of the boundary of a Gromov hyperbolic space X, one has a quasi-Moebius identification between the boundaries ∂ Y and ∂ X . For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the antipodal property. This gives a class of compact spaces called quasi-metric antipodal spaces. For any such space Z, we give a functorial construction of a boundary continuous Gromov hyperbolic space M (Z) together with a Moebius identification of its boundary with Z. The space M (Z) is maximal amongst all fillings of Z. These spaces M (Z) give in fact all examples of a natural class of spaces called maximal Gromov hyperbolic spaces. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called antipodal spaces and maximal Gromov product spaces. We prove that the injective hull of a Gromov product space X is isometric to the maximal Gromov product space M (Z) , where Z is the boundary of X. We also show that a Gromov product space is injective if and only if it is maximal. [ABSTRACT FROM AUTHOR]

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by h > 0 the mean curvature of horospheres in X, and set ρ = h / 2 . Fixing a basepoint o ∈ X , for ξ ∈ ∂ X , denote by B ξ the Busemann function at ξ such that B ξ (o) = 0 . Then for λ ∈ C the function e (i λ - ρ) B ξ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue - (λ 2 + ρ 2) . For a function f on X, we define the Fourier transform of f by f ~ (λ , ξ) : = ∫ X f (x) e (- i λ - ρ) B ξ (x) d v o l (x) for all λ ∈ C , ξ ∈ ∂ X for which the integral converges. We prove a Fourier inversion formula f (x) = C 0 ∫ 0 ∞ ∫ ∂ X f ~ (λ , ξ) e (i λ - ρ) B ξ (x) d λ o (ξ) | c (λ) | - 2 d λ for f ∈ C c ∞ (X) , where c is a certain function on R - { 0 } , λ o is the visibility measure on ∂ X with respect to the basepoint o ∈ X and C 0 > 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon. [ABSTRACT FROM AUTHOR]

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension of least three) for G1, G2 cocompact Kleinian groups, any quasiconformal map pairing a G1-invariant pattern to a G2-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math. GT/08094449, (2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G1 and a quasiconformal conjugate h-1G2h of a cocompact group G2. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence. [ABSTRACT FROM AUTHOR]