Jeremy Kahn

Hyperbolic Volume of Manifolds with Geodesic Boundary and Orthospectra

In this paper we describe a function Fn : R+ → R+ such that for any hyperbolic n-manifold M with totally geodesic boundary

Immersing almost geodesic surfaces in a closed hyperbolic three manifold

{\partial M \neq \emptyset}

The Quasi-Additivity Law in conformal geometry

, the volume of M is equal to the sum of the values of Fn on the orthospectrum of M. We derive an integral formula for Fn in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.

Local connectivity of Julia sets for unicritical polynomials

We study the affine orbifold laminations that were constructed by Lyubich and Minsky (J. Differential Geom. 47(1) (1997), pp. 17–94). An important question left open in the original construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen was itself a question raised by Lyubich and Minsky.