A nonlinear Poisson transform for Einstein metrics on product spaces

Abstract

Abstract. We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If M is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of M.

An extension problem for the CR fractional Laplacian

“…Scattering theory on asymptotically K-hyperbolic manifolds was developed in [7,59] (see also [11,60] for the generalization to differential forms). Here we would like to show that in the case of hyperbolic space H m Q , the calculation of the conformal fractional Laplacian P η κ from Theorem 1.1 and the energy identity from Theorem 1.2 are analogous.…”

Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces

“…We say that g is an asymptotically hyperbolic metric with conformal infinity [ g] if there exists ν > 0 and some k ∈ ρ ν C ℓ,α (M, T 2 M ), ℓ ≥ 2 so that in the product neighborhood g = g ⋆ + k. Such a metric need not be smoothly conformally compact. These spaces have been introduced by Biquard [3] and we use the analysis of the corresponding geometric Laplace operators that appears in [4].…”

“…Geometric background. In this section we define the class of asymptotically hyperbolic metrics of interest and refer the reader to [4] for further details. We begin by discussing the model metrics.…”

“…In [9,10], Mazzeo constructs a pseudodifferential calculus, the 0-calculus, that contains the inverses of operators like P g . We refer the reader to [4] for a concise treatment of the details; see also [3,7] for other approaches.…”

Fractional Laplacians and extension problems: The higher rank case

“…There are already several works dealing with the resolvent operator for (1.3) on higher rank symmetric spaces, and its geometric properties [47,3], especially in relation to quantum n-body particle scattering. In the specific product case, we look at the relation to the scattering operator constructed in [4,44,31], which does not satisfy the conformal covariance property but it has been an inspiration for our work. The difference from their construction to ours comes from the distinction between weakly and strongly harmonic functions (see, for instance, the survey [41]), and the difference between the Martin boundary and the geometric boundary.…”