Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space. Oblatum 2-IX-1994 & 7-VIII-1995     

Multipliers on rank one locally symmetric spaces

We study the L ^p -boundedness of spectral multipliers on non-compact locally symmetric spaces of rank one.     

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces , . For the quaternionic hyperbolic spaces , , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

     

On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the Sn and CPn cases, there are harmonic 2-tori into the quaternionic projective space HPn which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in CPn or Sn which is simultaneously of finite type and of finite uniton number must be constant.     

Ford fundamental domains in symmetric spaces of rank one

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.     

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

Fractional integrals on compact Riemannian symmetric spaces of rank one

In this paper we study mixed norm boundedness for fractional integrals related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.     

Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z ₂-coefficients.The second author was supported in part by FAPESP and CNPq.     

Bohr-Favard inequality for functions on compact symmetric spaces of rank one

Mathematical Notes -     

Marked length rigidity of rank one symmetric spaces and their product

In this paper we show that if two Zariski dense representations, from a group G into Iso(X) where X is rank one symmetric space, have the proportional marked length spectrum, then they are conjugate. As a generalization we show that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

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

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one

We study the problem of inversion of the local Pompeiu transform on Riemannian symmetric spaces of rank one. The explicit formula for the reconstruction of a function by its averages over balls and spheres with a single fixed radius is obtained.