Geometry Winter Workshop in Luxembourg

Abstract: It is a classical fact that any isometric embedding of the round sphere in the Euclidean space is obtained by a rigid motion. On the other hand it has been known since longtime that there are isometric embeddings of the hyperbolic plane in Minkowski space which are not umbilical. This phenomenon is related to the lack of compactness of the hyperbolic plane. Indeed the classification of properly embedded spacelike hyperbolic surfaces in Minkowski space relies on a notion of an asymptotic boundary, which can be regarded as a closed subset of the Penrose boundary of Minkowski space. In a work with Andrea Seppi and Peter Smillie we proved that there exists a bijective correspondence between asymptotic boundaries and properly embedded spacelike surfaces in Minkowski 3-space. However, typically those surfaces are not geodesically complete, so the space of embeddings of the hyperbolic plane in Minkowski space is still not understood. In a work in progress with Andrea Seppi and Peter Smillie we are investigating how the regularity of the boundary is related to the completeness of the surface. In the talk I will introduce the problem, explain the state of the art, and illustrate some recent results achieved with Andrea Seppi and Peter Smillie.

Abstract: Given a closed hyperbolic surface S, we consider quasi-Fuchsian manifolds homemorphic to SXR. We study their measured foliations at infinity which are the horizontal measured foliations of the Schwarzians derivatives obtained on the boundary at infinity by Uniformising their respective complex structures. We then discuss how given a pair of measured foliations (F,G) that fill a closed hyperbolic surface how tF and tG (where t>0 is small enough denotes the measure being scaled by a factor of t) can be realized as the measured foliations at infinity of a quasi-Fuchsian manifold which is sufficiently close to being Fuchsian. Starting from the definitions, the plan of the talk will be to give an overall picture of the setting of the theorem with the proof and see how it draws parallels to a similar result of Bonahon for bending measured laminations of a quasi-Fuchsian manifold.

Abstract: Let G/H be a homogeneous space where G and H are reductive Lie groups, both noncompact. For instance, G/H could be the pseudo-Riemannian hyperbolic space H^{p,q} = SO(p,q+1)/O(p,q) for p,q\geq 1. In order to understand compact manifolds locally modeled on G/H, we study properly discontinuous and cocompact actions of discrete groups on G/H. We prove the so-called Sharpness Conjecture: such actions always satisfy a strong form of properness called sharpness. As an application, we show that when the real corank of G/H is one (e.g. G/H = H^{p,q} with p>q), any discrete group Gamma admitting a properly discontinuous and cocompact action on G/H is Gromov hyperbolic, and the set of such actions is characterized by an Anosov representation condition, defining an open subset of Hom(Gamma,G). This is joint work with Nicolas Tholozan.

Abstract: Positivity is meant as a generalisation of the cyclic order on the circle and the notion of monotonicity a map from a cyclically ordered set in the circle.
Generalisation of the idea of positivity appeared to be crucial in understanding some connected components of the space of representations of a surface group in certain Lie groups G, although the common phenomenon was not figured out until recently.
In this talk, based on a preprint with Olivier Guichard and Anna Wienhard, I will start by examples generalizing this notion of cyclic order on the circle: convex curves and configurations in the plane, time like curve in Minkowski space. Then I will move to the general geometry of parabolic spaces and explain why the notion of positivity relates to special configurations of pairwise transverse triples and quadruples of points. This notion of positivity, which abides simple combinatorial properties, allows to define positive - or monotone - curves, then positive representations of surface groups. Positivity may not always exists and its occurences have been classified by Guichard—Wienhard.
I will then sketch the proof of our main result: positive representations are Anosov and fill up connected components of the space of representations.

Abstract: Abstract: We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n<=5. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic. The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

Abstract: Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of Θ-positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of Θ-positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.

Abstract: Recently, Markovic proved that there exists a maximal representation into (PSL(2,R))^3 such that the associated energy functional on Teichmuller space admits multiple critical points. That is, there is more than one minimal surface in the corresponding product of closed Riemann surfaces. We plan to discuss some aspects of the geometry of these minimal surfaces. We will explain that there is a relation to a notion of convexity for singular flat metrics, which has other interesting applications. This is work in progress, partially joint with Vladimir Markovic.

Abstract: In 2003 Donaldson applied infinite-dimensional symplectic reduction to prove the existence of a mapping class group invariant hyperKähler structure on an open neighbourhood of the Fuchsian locus in the space of quasi-Fuchsian manifolds. In this talk, I will discuss a similar question for the Lorenzian counterpart of quasi-Fuchsian manifolds, namely maximal globally hyperbolic Anti-de Sitter manifold, showing that the natural object that appears in this context is a para-hyperKähler structure. This structure recovers many of the previously known constructions, due to Mess, Krasnov-Schlenker, Bonsante-Mondello-Schlenker, and others. This is joint work with Filippo Mazzoli and Andrea Tamburelli.

Abstract: In 2013, Rossi proved that if a maximal globally hyperbolic (abbrev. MGH) conformally flat spacetime has two distinct homotopic lightlike geodesics with the same ends then it is a finite quotient of the Einstein universe. In this case, the ends of such lightlike geodesics are said to be conjugate. In the continuity of this result, I am interested in describing MGH conformally flat spacetimes with complete lightlike geodesics (i.e. which develop as lightlike geodesics joining two conjugate points in the Einstein universe). In this talk, I will describe an example that I call a Misner domain of the Einstein universe. Under some hypothesis, I prove that the universal covering of a MGH conformally flat spacetime with complete lightlike geodesics contains a Misner strip. The goal would be to prove that any MGH Cauchy compact conformally flat spacetime can be obtained by grafting (or removing) a Misner strip from another one. This would be the Lorentzian analogous of the operation of grafting on hyperbolic surfaces introduced by Thurston.

Abstract: A long standing question asks which reductive homogeneous spaces G/H admit a properly discontinuous and cocompact action of a discrete subgroup of G. The « sharpness » of such actions (see Fanny Kassel’s presentation) provides new obstructions to their existence, showing for instance that SL(n+1,R)/SL(n,R) does not have compact quotients for any n>2. A more refined topological obstruction, obtained jointly with Fanny Kassel et Yosuke Morita, gives the non-existence of compact quotients of many pseudo-hyperbolic spaces H(p,q).

Abstract: The 6-dimensional pseudosphere carries a natural almost-complex structure that is preserved by the action of the exceptional split real Lie group G_2. In this talk, we will describe the moduli space of holomorphic disks in the pseudosphere that are equivariant under the action of a surface group. This is a joint work with Brian Collier.