arXiv research

The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.

The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking …

Max flow/min cut theorem extended to currents and topology.

We prove quantitative recurrence and large deviations results for the Teichmuller geodesci flow on a connected component of a stratum of the moduli space $Q_g$ of holomorphic unit-area quadratic differentials on a compact genus $g \geq 2$ surface.

Superdense flows on surfaces imply bounded geodesics, and vice versa.

Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.

We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes th…

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.

Closed geodesics densely cover a circle in dilation surfaces.

Continuity of earthquake flow map transfers Teichmüller dynamics results.

We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geom…

We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in…

The Teichmüller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomo…

Study $\mathrm{SL}_2(\mathbb{R})$ dynamics on one-holed tori moduli space.

We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichmüller space and is…

In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manif…

Harmonic maps link Teichmüller spaces to framed representations.

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüll…

Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to b…

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a …

The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [1…

This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.

Flow maps into minimal surfaces with free boundary.

This text is an introduction to dilation surfaces. We attempt to expose some geometric and dynamical aspects of the subject: moduli spaces, directional foliations and the Teichmüller flow.

We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…

The Teichmuller unipotent flow can be defined concretely on certain moduli spaces of singular flat surfaces by shearing polygonal presentations of the surfaces. Thurston's earthquake flow on moduli spaces of hyperbolic surfaces is more mysterious. Both flows have deep and important connections to other areas of mathema…

Study connects landslide flow to integrable systems for harmonic maps.

Proves existence of unique circle packings on polyhedral surfaces.

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…

Study shows infinite volumes of moduli spaces for certain groups.

New triangulations encode flows with vanishing polynomial.

We consider the $SL(2,R)$ action on moduli spaces of quadratic differentials. If $μ$ is an $SL(2,R)$-invariant probability measure, crucial information about the associated representation on $L^2(μ)$ (and in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded …

The paper studies Hamiltonian flows for pseudo-Anosov mapping classes on surfaces.

We prove that for every flat surface $ω$, the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from $ω$ exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than $1$. This theorem extends a result by Chaika and Eskin whe…

The paper proves the existence of a unique circle packing on hyperbolic surfaces.

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmüller flow. The second one is more recent and uses a "labeling" of the underlying intervals, and was used in the proof of some recent major results a…

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

The Dehornoy order on braid groups is derived from a cluster algebra.

Rare Teichmüller disks converge to small limit sets.

The study examines the asymptotic behavior of extremal length in Teichmüller space.

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov …

We prove that the every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.

Study of Veech surfaces and their twist tori on moduli spaces of abelian differentials.

Study on Teichmüller rays' asymptotic behavior and distances.

We prove that the Bergman and the Teichmuller metrics are equivalent on Teichmuller spaces.