18 april 2005, kl 13.15-14.15, KTH, Seminarierum 3721.

Andreas Juhl (Humboldt-University, Berlin): Automorphic distributions, Selberg zeta functions and conformal geometry

Abstract. Kleinian groups are discrete subgroups of the isometry group of $n$-dimensional real hyperbolic space $H^n$. Any Kleinian group $\Gamma$ gives rise to a hyperbolic manifold $X = \Gamma \backslash H^n$ and a Selberg type zeta function $Z_\Gamma$ which encodes the periodic orbits structure of the geodesic flow of $X$. For convex-cocompact $\Gamma$ the zeta function $Z_\Gamma$ is meromorphic on the complex plane.A basic question is to study its zeros (and poles). $Z_\Gamma$ has a zero if there exists a distribution on the geodesic boundary $S^{n-1}$ of $H^n$ which is supported on the limit set of $\Gamma$, and is conformally invariant under $\Gamma$ with an exponent given by the zero. These distributions are called automorphic. The most prominent example is the Patterson-Sullivan measure which corresponds to a zero of $Z_\Gamma$ at the critical exponent (Hausdorff-dimension). The structure of automorphic distributions is very complex even in simple cases. Their study is naturally linked with various aspects of conformal geometry and quantum field theory: conformally invariant differential operators (generalized Yamabe operators), Branson's $Q$-curvature and the AdS/CFT correspondence of Maldacena. We describe recent results on polynomial families of (generalized) Yamabe operators which are canonically associated to conformal submanifolds, and their roles in the above mentioned contexts. This is an area full of challenging problems.