Speaker
Zuo Lin, University of California, Berkeley
Abstract
An important theme in geometry and dynamics is the behavior of (totally geodesic) hyperplanes in spaces of negative curvature. Here by hyperplane we mean totally geodesic codimension one submanifold. More precisely, let M be a (d + 1)-manifold with negative sectional curvature with d ≥ 2, one can ask the following two questions.
(1) How many hyperplanes with finite d-volume are there in M?
(2) If there exists such a hyperplane P with finite d-volume, is there any restriction on vol_d(P) from the geometry of the ambient manifold M?
In this talk, I will present a recent joint work with Pratyush Sarkar and Liyang Shao providing effective upper bounds to both questions in the case where M is an infinite volume geometrically finite hyperbolic manifold. Our bounds depend only on simple geometric quantities of M: critical exponent, volume of convex core, and injectivity radius. The proof of the upper bounds utilizes ideas from recent progress on an effective version of Ratner’s theorem by Lindenstrauss-Mohammadi-Wang-Yang and spectral aspects of Patterson-Sullivan theory.