# Workshop (speakers and titles)

Workshop, June 27 to July 1

Speakers and Titles

Igor Belegradek Georgia Institute of Technology

Title: Smoothness of Minkowski sum and generic rotations (with Zixin Jiang)

Abstract: I will discuss whether the Minkowski sum of two compact convex bodies can be made smoother by a generic rotation of one of them.  Here "generic" is understood in the sense of Baire category. The main result is a construction of an infinitely differentiable convex plane domain whose Minkowski sum with any generically rotated copy of itself is not five times differentiable.

Reto Buzano Queen Mary University of London

Title: Minimal hypersurfaces with bounded index and bounded area

Abstract: We study sequences of closed minimal hypersurfaces (in closed Riemannian manifolds) that have uniformly bounded index and area. In particular, we develop a bubbling result which yields a bound on the total curvature along the sequence. As a consequence, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area. This is joint work with Ben Sharp.

David Gabai Princeton University

Title:  Maximal cusps of low volume (with Robert Haraway, Robert Meyerhoff, Nathaniel Thurston and Andrew Yarmola)

Abstract:  We address the following question. What are all the 1-cusped hyperbolic 3-manifolds whose maximal cusps have low volume?  Among other things we will outline a proof that the figure-8 knot complement and its sister are the 1-cusped manifolds with minimal maximal cusp volume.

Robert Haslhofer University of Toronto

Title: The moduli space of 2-convex embedded spheres

Abstract: We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture, M_2 is contractible. This is a highly nontrivial theorem generalizing in particular the Schoenflies theorem and Cerf’s theorem.

In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants.I will start by sketching a proof of Smale’s theorem that M_1 is contractible. By a beautiful theorem of Grayson, the curve shortening flow deforms any closed embedded curve in the plane to a round circle, and thus gives a geometric analytic proof of the fact that M_1 is path-connected. By flowing, roughly speaking, all curves simultaneously, one can improve path-connectedness to contractibility.In the second half of my talk, I’ll describe recent work on space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery.

Sa'ar Hersonsky University of Georgia

Title: Electrical Networks and Stephenson's Conjecture

Abstract: The Riemann Mapping Theorem asserts that any simply connected planar domain which is not the whole of it, can be mapped by a conformal homeomorphism onto the open unit disk. After normalization, this map is unique and is called the Riemann mapping. In the 90's, Ken Stephenson, motivated by a circle packing approximation scheme suggested by Thurston (and first proved to converge by Rodin-Sullivan), predicted that the Riemann Mapping may be approximated by a different scheme, i.e., by a sequence of finite networks endowed with particular choices of conductance constants. These networks are naturally defined in terms of the contact graph of any circle packing.

We will affirm Stephenson's Conjecture in a greater generality.

Vitali Kapovitch University of Toronto

Title: On noncollapsed almost Ricci-flat 4-manifolds

Abstract: We show that a non-collapsed almost Ricci-flat spin 4-manifold with nonzero $\hat{A}$-genus must be diffeomorphic to a K3 surface. This is joint work with John Lott.

Daniel Ketover Imperial College London

Title: Sharp entropy bounds of closed surfaces and min-max theory

Abstract: In 2012, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface in R^3 is at least that of the self-shrinking two-sphere. I will explain joint work with X. Zhou where we interpret this conjecture as a parabolic version of the Willmore problem and give a min-max proof of (most cases) of their conjecture.

Feng Luo Rutgers University

Title: Discrete conformal geometry of polyhedral surfaces and its convergence

Abstract: Our recent joint work with D. Gu established a discrete version of the uniformization theorem for compact polyhedral surfaces.   In this talk, we prove that discrete uniformizaton maps converge to conformal maps when the triangulations are sufficiently fine chosen.  We will also discuss the relationship between the discrete uniformization theorem and convex polyhedral surfaces  in the hyperbolic 3-space.  This is a joint work with J. Sun and T. Wu.

Vladimir Markovic California Institute of Technology

Title: Harmonic quasi-isometries between negatively curved manifolds

Abstract: Very recently, Markovic, Lemm-Markovic and Benoist-Hulin, established the existence of a harmonic mapping in the homotopy class of an arbitrary quasi-isometry between rank 1 symmetric spaces. I will discuss these results and the more general conjecture which states that this result holds for quasi-isometries between negatively curved manifolds and metric spaces.

Laurent Mazet Université Paris-Est - Créteil & CNRS

Title: Minimal hypersurfaces of least area

Abstract: In this talk, I will present a joint work with H. Rosenberg where we give a characterization of the minimal hypersurface of least area in any Riemannian manifold. As a consequence, we give a lower bound for the area of a minimal surface in a hyperbolic 3-manifold.

Melanie Rupflin University of Oxford

Title: Horizontal curves of metrics and applications to geometric flows

Abstract: On closed surfaces there are three basic ways to evolve a metric, by conformal change, by pull-back with diffeomorphisms and by horizontal curves, moving orthogonally to the first two types of evolution. As we will discuss in this talk, horizontal curves are very well behaved even if the underlying conformal structures degenerate in moduli space as $t\to T$. We can describe where the metrics will have essentially settled down to the limit by time $t<T$ as opposed to regions on which the metric still has to do an infinite amount of stretching. This quantified information is essential in applications and allows us to prove a "no-loss-of-topology" result at finite time singularities of Teichm\"uller harmonic map flow which, combined with earlier work, yields that this geometric flow decomposes every map into a collection of branched minimal immersions and curves.

This is joint work with Peter Topping.

Gregory McShane Université Grenoble Alpes

Title: Volumes of hyperbolics manifolds and translation distances

Abstract: Schlenker and Krasnov have established a remarkable Schlaffli-type formula for the (renormalized) volume of a quasi-Fuchsian manifold. Using this, some classical results in complex analysis and Gromov-Hausdorff  convergence for sequences of open 3-manifolds due to Brock-Bromberg one obtains explicit upper bounds for the volume of a mapping torus in terms of the translation distance of the monodromy on Teichmueller space. We will explain Brock-Bromberg's approach to the Thurston's uniformization theorem for hyperbolic manifolds which are mapping tori. In particular the "coarse geometry" of the convex core of a quasi fuchsian manifold.

This is joint work with S.Kojima.

Stéphane Sabourau Université Paris-Est - Créteil

Title: Sweep-outs, width estimates and volume

Abstract: Sweep-out techniques in geometry and topology have recently received a great deal of attention, leading to major breakthroughs. In this talk, we will present several width estimates relying on min-max arguments in relation to the volume of Riemannian manifolds. Dealing with the case of surfaces first, we will focus our attention on generalisations in higher dimension and present new estimates obtained in a work in progress.

Jean-Marc Schlenker Université du Luxembourg

Title: Anti-de Sitter geometry and polyhedra inscribed in quadrics

Abstract: Anti-de Sitter geometry is a Lorentzian analog of hyperbolic geometry. In the last 25 years a number of connections have emerged between 3-dimensional anti-de Sitter geometry and the geometry of hyperbolic sufaces. We will explain how the study of ideal polyhedra in anti-de Sitter space leads to an answer to a question of Steiner (1832) on the combinatorics of polyhedra that can be inscribed in a quadric. Joint work with Jeff Danciger and Sara Maloni.

Juan Souto Université de Rennes 1

Title: Counting curves on surfaces

Abstract: An old theorem of Huber asserts that the number of closed geodesics of length at most L on a hyperbolic surface is asymptotic to $\frac{e^L}L$. However, things are less clear if one either fixes the type of the curve, possibly changing the notion of length, or if one counts types of curves. Here, two curves are of the same type if they differ by a mapping class. I will describe some results in these directions.

Jeff Viaclovsky University of Wisconsin

Title: Deformation theory of scalar-flat Kahler ALE surfaces

Abstract: I will discuss a Kuranishi-type theorem for deformations of complex structure on ALE Kahler surfaces, which will be used to prove that for any scalar-flat Kahler ALE surface, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics can then be constructed, which is universal up to small diffeomorphisms. I will also discuss a formula for the dimension of the local moduli space in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of an isolated quotient singularity.  This is joint work with Jiyuan Han.

Genevieve Walsh Tufts University

Title: Boundaries of Kleinian groups

Abstract: We study the problem of classifying Kleinian groups via the topology of their limit sets. In particular, we are interested in one-ended convex-cocompact Kleinian groups where each piece in the JSJ decomposition is a free group, and we describe interesting examples in this situation.  In certain cases we show that the type of Kleinian group is determined by the topology of its group boundary.  We conjecture that this is not the case in general.  We also determine the homeomorphism types of planar boundaries that can occur.  This is joint work in progress with Peter Haissinsky and Luisa Paoluzzi.

Burkhard Wilking Universität Münster

Title: Manifolds with almost nonnegative curvature operator

Abstract: We show that n-manifolds with a lower volume bound v and upper diameter bound D whose curvature operator is bounded below by $-\varepsilon(n,v,D)$ also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound. nonnegative curvature operator.

Robert Young New York University

Title: Quantitative rectifiability and differentiation in the Heisenberg group

Abstract: (joint work with Assaf Naor) The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is unusually difficult to embed in $\mathbb{R}^n$. Cheeger and Kleiner introduced a new notion of differentiation that they used to show that it does not embed nicely into $L_1$. This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques that let us quantify the "roughness" of such surfaces, find sharp bounds on the distortion of embeddings of $\mathbb{H}$, and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem.

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