Below are some seminars that I have organized at SIMIS.
Speaker: Martin Skandar Chuaqui from Pontificia Universidad Católica de Chile
Title: Envelopes of horospheres in hyperbolic space
Abstract: We present the main ideas of the work by Ch. Epstein on envelopes of a family of horospheres in hyperbolic n-space determined by a support function on a domain on the asymptotic boundary. We present several important formulas associated with this construction, crucially, how the principal curvatures of the envelope evolve under the parallel flow.
Title: The Schwarzian and quasiconformal reflections on the n-sphere
Abstract: We present briefly main elements of the work of Osgood and Stowe on the Schwarzian derivative for conformal mappings between Riemannian manifolds and focus on an injectivity criterion for such mappings when the target is the n-sphere. We show how to formulate the injectivity criterion as a condition for a conformal metric on the image domain on the n-sphere, and an associated support function that induces an envelope with stable principal curvatures under the parallel flow.
Title: Renormalized Volume for Hyperbolic manifolds I, II
Martin Bridgeman from Boston College
Abstract: Renormalized volume was first introduced by Graham-Witten for conformally compact Einstein manifolds and is closely related to the study of Liouville action by Taktajan-Zograf and Taktajan-Teo and others. Krasnov-Schlenker developed the theory in the hyperbolic setting, proving a number of important results which relate renormalized volume to convex core volume and Weil-Petersson geometry. This new tool to study hyperbolic manifolds has the potential to unify many strands of the theory as well as provide new insights. We will begin with an introduction to the theory, outlining the work of Krasnov-Schlenker. For N a compact hyperbolizable 3-manifold, renormalized volume gives an analytic function V_R on the space CC(N) of convex compact hyperbolic structures.
We will then introduce the (Weil-Petersson) gradient flow of V_R on CC(N) and show that it gives a geometric flow which attempts to uniformize the boundary and minimize the volume. In particular we will discuss the result that for N acylindrical the flow has a global attractor at the unique structure M_g which has totally geodesic convex core boundary. Time permitting, we will discuss generalizations to Universal Weil-Petersson space T_0(1) where we show that the Weil-Petersson gradient of Lowner energy has a global fixed point at the geometric circle and thus gives a curve straightening flow in this case.