arXiv subjects
Publications and source records attributed to François Labourie.
In the case of the complex hyperbolic space, we establish a version of the hyperbolic Ax--Lindemann Theorem for the topological closure rather than the Zariski closure. Our approach combines classical differential geometry with homogeneous dynamics.
We investigate and define in this paper, in the context of the correspondence between anti-de Sitter $3$-space and $(1,1)$-conformal metrics, the analogs of $\cW$-volume, Epstein surfaces, and Liouville action. These notions were well-studied in the correspondence between $3d$-hyperbolic manifolds and $2d$ conformal metrics. We apply our construction to positive curves in flag manifolds equipped with a positive structure to obtain invariants of these curves that are finite in the case of piecewise circles.
The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions and, correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal algebra called {\em ghost algebra} related in some cases to the swapping algebra introduced by the second author. As a consequence of our main theorem, we show that the set of those functions -- length and correlation -- is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichmüller space, secondly we exhibit subalgebras of commuting functions. An important tool is the study of {\em uniformly hyperbolic bundles} which is a generalisation of Anosov representations beyond periodicity.
In this note, we answer positively a question of Yau by proving the existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension. The proof follows the strategy of Song, applying it to closed Riemann surfaces with large automorphism groups, and obtaining almost hyperbolic minimal surfaces.
In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Θ$-positivity, a generalization of Lusztig's total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $Θ$-positive representations of surface groups. We prove that $Θ$-positive representations are $Θ$-Anosov. This implies that $Θ$-positive representations are discrete and faithful and that the set of $Θ$-positive representations is open in the representation variety. We show that the set of $Θ$-positive representations is closed within the set of representations that do not virtually factor through a parabolic subgroup. From this we deduce that for any simple Lie group $\mathsf G$ admitting a $Θ$-positive structure there exist components consisting of $Θ$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Θ$-positive representations.
We prove that $Θ$-positive representations of fundamental groups of surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and their associated cross-ratios are positive. As a consequence we deduce that $Θ$-positive representations form closed subsets of the representation variety.
We give a Chern-Simons based proof of Tholozan's volume formula for closed anti-de-Sitter 3-manifolds.
We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the pseudo-hyperbolic space which are limits of positive curves. We also discuss a compact Plateau problem. The required compactness arguments rely on an analysis of the pseudo-holomorphic curves defined by the Gauss lifts of the maximal surfaces.
We study in this paper quasiperiodic maximal surfaces in pseudo-hyperbolic spaces and show that they are characterised by a curvature condition, Gromov hyperbolicity or conformal hyperbolicity. We show that the limit curves of these surfaces in the Einstein Universe admits a canonical quasisymmetric parametrisation, while conversely every quasisymmetric curve in the Einstein Universe bounds a quasiperiodic surface in such a way that the quasisymmetric parametrisation is a continuous extension of the uniformisation; we give applications of these results to asymptotically hyperbolic surfaces, rigidity of Anosov representations and a version of the universal Teichmüller space.
A report on an article of Calegari, Marques and Neves about counting minimal surfaces. An idea of a proof is included using the concept of a laminar measure.
We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are Hölder. Using this notion, we show a quantitative version of our surface subgroup theorem and in particular that one can obtain $K$-Sullivan limit maps, as close as one wants to smooth round circles. All these results use the coarse geometry of "path of triangles" in a certain flag manifold and we prove an analogue to the Morse Lemma for quasi-geodesics in that context.
We give an independent proof of a theorem of Danciger of Zhang: surface groups with Hitchin linear part cannot act properly on the affine space
We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in the proof, we establish a transversality result for positive quadruples of flags.
We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.
We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\mathsf{SL}_n(\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symplectic structure and to the Drinfel'd--Sokolov reduction. We also prove an extension of Wolpert formula.
The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations into PSL(n,R) along the Fuchsian locus. The expression is in terms of a parametrization of the tangent space by holomorphic differentials, and it gives a precise relationship with the Petersson pairing. Along the way, variational formulas are established that generalize results from classical Teichmueller theory, such as Gardiner's formula, the relationship between length functions and Fenchel-Nielsen deformations, and variations of cross ratios.
We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of $\mathsf G_0$. Some partial extensions of the construction hold for cyclic bundles in higher rank.
These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.