ICTS

Neck-pinching of CP^1-structures

A CP^1-structure on a surface is the geometric structure modeled on the Riemann sphere, and it’s holonomy representation is a homomorphism from the fundamental group of the surface into PSL(2, C). We discuss about certain degenerations of CP^1-structures when their holonomy representations converge and their conformal structures are pinched along loops.
 

Hyperbolic p-barycenters, circumcenters, and Moebius maps

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between the boundaries of proper, geodesically complete CAT(-1) spaces, we define an extension $F : X \to Y$ of $f$, called the circumcenter extension of $f$, which is shown to be a $(1, \log 2)$-quasi-isometry, which is locally $1/2$-Holder continuous. If $X, Y$ are complete, simply connected Riemannian manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$, then the circumcenter extension is a $(1, (1 - 1/b) \log 2)$-quasi-isometry. Moreover if $g : \partial Y \to \partial X$ is the inverse of $f$, then the circumcenter extensions $F : X \to Y$ and $G : Y \to X$ of $f$ and $g$ are inverses of each other, and are $\sqrt{b}$-bi-Lipschitz. This is proved by constructing a family of extensions $F_p : X \to Y, 1 \leq p \leq \infty$ of $f$, called the hyperbolic $p$-barycenter extensions of $f$, and analyzing their behaviour as $p$ tends to $\infty$. As a corollary we obtain that if two closed, negatively curved manifolds have the same marked length spectrum, then they are bi-Lipschitz homeomorphic.
 

Branched Holomorphic Cartan Geometries and Calab i-Yau manifolds

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. (Joint work with Sorin Dumitrescu.).
 

Limits of cubic differentials and projective structures

A construction due independently to Labourie and Loftin identifies the space of convex RP^2 structures on a compact surface with the bundle of holomorphic cubic differentials over the moduli space of Riemann surfaces. We study pointed geometric limits of sequences that go to infinity in the space of RP^2 structures while remaining over a compact set in moduli space. For such a sequence, we construct a local limit polynomial (in one complex variable) which describes the rate and direction of accumulation of zeros of the cubic differentials about the sequence of base points. We then show that this polynomial determines the convex polygon in RP^2 that is the geometric limit of the images of the developing maps of the projective structures. This is joint work with Michael Wolf.
 

Special points in the character variety

We discuss Anosov representations of a surface group into a simple Lie group G of non-compact type and show that for every cocompact lattice L in G, there exists an Anosov representation of a surface group with image in L.
 

Convex real projective Dehn fillings

Thurston's hyperbolic Dehn surgery theorem says that if M is a cusped hyperbolic three dimensional manifold then almost all Dehn fillings of M admit a hyperbolic structure. However, the hyperbolic Dehn filling is impossible for dimension bigger than three. In this talk, I will give the first examples of cusped hyperbolic four dimensional manifolds whose Dehn fillings admit a convex real projective structure. Joint work with Suhyoung Choi and Ludovic Marquis.
 

On cyclic Higgs bundles

We derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of f. As an application, we give a complete picture for maximal Sp(4,R)-representations in the 2g−3 Gothen components and the Hitchin components. This is joint work with Song Dai.
 

Algebraic Ending Laminations and Quasiconvexity

We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence 1→H→G→Q→1 of hyperbolic groups. These laminations arise in different contexts: existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on $\R-$trees. We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite index subgroups of H, the normal subgroup in the exact sequence above.This is joint work with Kasra Rafi.
 

Complex hyperbolic representations of triangle groups

The holomorphic isometry group of complex hyperbolic space is PU(n,1). In this talk I will discuss representations of triangle groups in PU(2,1). Schwartz gave a conjectural picture of the situation when the reflections in the sides of the triangle have order 2. Some of his conjectures have been proved (or disproved) but most remain open. The situation is more complicated when the reflections have higher order (which can happen for complex reflections). I will give a survey of recent work in this area and discuss some open problems.
 

On Ultra-Parallel Complex Hyperbolic Triangle Groups

Ultra-parallel complex hyperbolic triangle groups are groups of isometries of the complex hyperbolic plane generated by 3 complex reflections in complex geodesics which do not intersect pair-wise. I will discuss discreteness and non-discreteness results for certain types of such groups. This is joint work with A. Monaghan and J. Parker and with S. Povall.
 

Geodesic currents and counting problems

We show that, for every filling geodesic current, a certain scaled average of the mapping class group orbit of this current converges to multiple of the Thurston measure on the space of measured laminations. This has applications to several counting problems, in particular, in counting the number of closed curves on a surface and counting the number of lattice points in the ball of radius R in Teichmüller space equipped with Thurston’s asymmetric metric. This is a joint work with Juan Souto.
 

Sheared Pleated surfaces and Limiting Configurations for Hitchin's equations

A recent work by Mazzeo-Swoboda-Weiss-Witt describes a stratum of the frontier of the space of SL(2,C) surface group representations in terms of 'limiting configurations' which solve a degenerated version of Hitchin's equations on a Riemann surface. We interpret these objects in (a mapping class group invariant way in) terms of the hyperbolic geometric objects of shearings of pleated surfaces. The effect is to refine the Morgan-Shalen compactification for this stratum. Much of the talk is devoted to defining the various objects. (Joint with Andreas Ott, Jan Swoboda, and Richard Wentworth).
 

The Goldman symplectic form on the Hitchin component

Let S be a closed, orientable, connected surface of genus at least 2. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us to find a maximal family of Poisson commuting Hamiltonian flows on the Hitchin component. This generalizes the well-known fact that on Teichmüller space, the twist flows along a pants decomposition of S is a maximal family of Poisson commuting Hamiltonian flows. This is joint work with Zhe Sun and Anna Wienhard.