Hyperbolic manifolds

Abstract: We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will talk about some results concerning the growth of L2 norm/Thurston norm for a sequence of closed hyperbolic 3-manifolds converging geometrically to a cusped manifold, using Dehn filling and minimal surface.

Abstract: I will start by giving a brief overview of quasifuchsian manifolds and then talk about work Uhlenbeck did on parametrizing a subset of quasifuchsian manifolds-- called almost fuchsian manifolds-- by their minimal surfaces.  In the second part I will talk about a compactification of the space of almost Fuchsian manifolds that Zeno Huang and myself constructed, and give some applications.  We are able to prove for example that there is a universal \epsilon depending only on g such that for every closed hyperbolic 3-manifold that fibers over the circle with fiber a surface of genus g, every stable minimal surface isotopic to the fiber has a principal curvature larger than 1+\epsilon somewhere.  (I should also have mentioned in my talk, but forgot so I will mention here, that James Farre and Franco Vargas Pallete prove similar results, but by different methods, in a recent paper they wrote.)  

Abstract: In a recent work Bridgeman, Brock, and Bromberg characterized the infimum of the renormalized volume of a convex co-compact hyperbolic 3-manifold with incompressible boundary, as we deform its structure by quasi-isometries. In this talk, I will describe a series of similarities between the renormalized volume and another notion of volume for such a class of manifolds, namely the dual volume of the convex core. In particular, we will see how these analogies and the properties of Gaussian curvature surfaces allow us to obtain a similar characterization for the dual volume function.

Abstract: We define a special class of infinite volume hyperbolic 3-manifolds that contain unique minimal surfaces and study the global geometry of their deformation space. We then consider the analogous picture in the context of Lorentzian geometry. We will see that these manifolds share many similarities with their hyperbolic counterparts and are in some sense even better behaved. We then describe the global parahyperkahler geometry of their deformation space. 

Abstract: In this talk I will give a quick overview of spectral geometry and describe how it relates to hyperbolic geometry by outlining some explicit connections. I will then describe how isoperimetric ratios can be used to give geometric interpretations of spectral gaps. Ultimately, I aim to describe a new isoperimetric ratio that controls the coexact spectral gap in hyperbolic manifolds.  

Abstract: The deformation space of a manifold is a neighbourhood of hyperbolic structures that it admits. We will focus here on the parametrization of the deformation space through ideal triangulations. In the first talk we will cover the case where the manifold is orientable. In the second one we will generalize the results to the non-orientable case, as well as discuss the method of computing deformations through the variety of representations.

Abstract: What are the accumulation points of the set of dilatations corresponding to a fibered cone in a hyperbolic 3-manifold? The main theorem of this talk is that, other than perhaps 1, they can all be interpreted as stretch factors of automorphisms of infinite type surfaces arising as noncompact leaves of depth one foliations. The theorem is an application of results in a forthcoming preprint, which is joint with Yair Minsky and Samuel Taylor.

Abstract: Geometric finiteness was originally introduced by Ahlfors for discrete isometry subgroup of 3-dimensional hyperbolic space. It was later generalized to higher dimensional hyperbolic spaces and negatively pinched Hadamard manifolds by Bowditch. In this talk, we will review the definitions and give new characterization of Geometric finiteness in negatively pinched Hadamard manifolds.  

The canonical lift of a closed geodesic a hyperbolic surface is a link in the unit tangent bundle. Drilling this link produces a 3-manifold. When the geodesic filling with primitive components Foulon-Hasselblatt (2013) proved that it is a hyperbolic 3-manifold. Upper and lower bounds for the volume of this manifold in terms of geometric invariants of the multicurve have been studied in recent literature. For example, Rodriguez-Migueles (2020) gives a lower bound of the volume in terms of the number of homotopy classes of subarcs of the curve between boundaries of a pants decomposition.

In this talk we let the closed geodesic be a random geodesic, i.e., the closed geodesic depending on T obtained by pulling tight the curve obtained by following a dense trajectory of the geodesic flow for time T and then closing off. We use the topological lower bound and exponential mixing of geodesic flow to give a lower bound sublinear in T for the volume of the corresponding 3-manifold.

This is joint work with Tommaso Cremaschi, Yannick Krifka and Franco Vargas Pallete.

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them.  In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic.  In this talk, we will discuss an infinite family of twist knot complements containing exactly 1 totally geodesic surface, and how to use covering space theory to extend this result to present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly k totally geodesic surfaces for every positive integer k.  This is joint work with Khanh Le.

Abstract: In this pair of talks we explore the homeomorphism problem for hyperbolic manifolds. In particular we look at what it might take to find a bounded runtime algorithm to compare finite volume hyperbolic n-manifolds (for n>2). In the first talk we'll see where approaches using Gromov Hyperbolicity and Manifold Theory succeed, and why they might fail to give us bounded runtime.  In the second talk, we'll look at using systems of polynomials to control the geometry of hyperbolic manifolds and how that ties in to the Homeomorphism Problem.