Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

2000 ◽  
Vol 10 (2) ◽  
pp. 171-196
Author(s):  
Yuguang Shi ◽  
You-De Wang Not Available
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Boundary Value ◽  
Hyperbolic Spaces ◽  
Singular Boundary ◽  
Rank One ◽  
Complex Hyperbolic Spaces

scholarly journals Harmonic maps on hyperbolic spaces with singular boundary value

1999 ◽  
Vol 51 (3) ◽  
pp. 551-600 ◽  
Author(s):  
Yuguang Shi ◽  
Luen-Fai Tam ◽  
Tom Y.-H. Wan
Keyword(s):  
Harmonic Maps ◽  
Boundary Value ◽  
Hyperbolic Spaces ◽  
Singular Boundary

scholarly journals McShane's identity in rank one symmetric spaces

2014 ◽  
Vol 157 (1) ◽  
pp. 113-137
Author(s):  
INKANG KIM ◽  
JOONHYUNG KIM ◽  
SER PEOW TAN
Keyword(s):  
Symmetric Spaces ◽  
Hyperbolic Spaces ◽  
Surface Groups ◽  
Isometry Groups ◽  
Rank One ◽  
Complex Hyperbolic Spaces

AbstractWe study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this class of identities.

scholarly journals An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

2021 ◽  
Author(s):  
Laura Atanasi ◽  
Massimo A. Picardello
Keyword(s):  
Harmonic Functions ◽  
Symmetric Spaces ◽  
Cartesian Product ◽  
Measurable Subset ◽  
Hyperbolic Spaces ◽  
Area Theorem ◽  
Area Function ◽  
Homogeneous Trees ◽  
Rank One ◽  
Higher Rank

AbstractFor harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Harmonic maps from surfaces into pseudo-Riemannian spheres and hyperbolic spaces

1983 ◽  
Vol 94 (3) ◽  
pp. 483-494 ◽  
Author(s):  
S. Erdem
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Hyperbolic Spaces ◽  
Euclidean Sphere ◽  
Holomorphic Maps ◽  
Space Forms ◽  
Indefinite Metrics ◽  
Complex Hyperbolic Spaces ◽  
Complex Projective

In [2, 4, 5, 6, 7] Calabi, Barbosa and Chern showedthat there is a 2:1 correspondence between arbitrary pairs of full isotropic (terminology as in [8]) harmonic maps ±φ:M→S2mfrom a Riemann surface to a Euclidean sphere and full totally isotropic holomorphic maps f:M→2mfrom the surface to complex projective space. In this paper we show, very explicitly, how to construct a similar one-to-one correspondence whenS2mis replaced by some other space forms of positive and negative curvatures with their standard (indefinite) metrics obtained by restricting a standard (indefinite) bilinear form on Euclidean space to the tangent spaces. We get over a difficulty encountered by Barbosa of dealing with the zeros of a certain wedge product by a technique adapted from [8]. The case of complex projective space forms (indefinite complex projective and complex hyperbolic spaces) will be considered in a separate paper. Some further developments in classification theorems are given by Eells and Wood [8], Rawnsley[14], [15] and Erdem and Wood [10].

scholarly journals Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

2021 ◽  
Vol 2021 (779) ◽  
pp. 189-222
Author(s):  
José Carlos Díaz-Ramos ◽  
Miguel Domínguez-Vázquez ◽  
Alberto Rodríguez-Vázquez
Keyword(s):  
Symmetric Spaces ◽  
Hyperbolic Spaces ◽  
Orbit Equivalence ◽  
Principal Curvatures ◽  
Cohomogeneity One ◽  
Isoparametric Hypersurfaces ◽  
Rank One ◽  
Constant Principal Curvatures

Abstract We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

scholarly journals Resonances and Scattering Poles in Symmetric Spaces of Rank One

2018 ◽  
Vol 2019 (20) ◽  
pp. 6362-6389
Author(s):  
Sönke Hansen ◽  
Joachim Hilgert ◽  
Aprameyan Parthasarathy
Keyword(s):  
Laplace Operator ◽  
Half Plane ◽  
Symmetric Spaces ◽  
Boundary Value ◽  
Square Root ◽  
Boundary Values ◽  
Resolvent Set ◽  
Rank One ◽  
Riemannian Symmetric Spaces ◽  
The Laplace Operator

Abstract We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of Kashiwara–Ōshima. From this, we derive that the poles of the corresponding meromorphic continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp–Stein intertwiner. As a by-product of the underlying ideas, we obtain a new and self-contained proof of Helgason’s conjecture for distributions in the case of rank one symmetric spaces.

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