Geometry and Topology Seminar: New results about noncompact harmonic and asymptotically harmonic spaces.
12 February 2015 13:00 in CM221
Harmonic and asymptotically harmonic spaces are Riemannian manifolds where the
Laplace-Beltrami operator assumes a particularly simple form in polar coordinates and horocyclic coordinates,
respectively. Known noncompact examples are, besides the Euclidean spaces, all rank one symmetric spaces and Damek-Ricci spaces. Useful tools in the treatment of these spaces come from Riemannian and spectral geometry. In this talk, we will discuss recent results based on spherical and horocyclic averages which were obtained in collaboration with Evangelia Samiou (in the case of harmonic spaces) and Gerhard Knieper (in the case of asymtptotically harmonic spaces). It should be mentioned that the most challenging problem in this area is
the classification of these spaces in the noncompact case which is still widely open. (The classification problem in the compact case, known as "Lichnerowicz conjecture", was settled in 1990 by Z.I. Szabo.).