Cyclic surfaces and Hitchin components in rank 2

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 Teichm\"uller 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.Comment: 61 pages v3. Final version, with more typos corrected as well as the statement of Proposition 6.3.1 (cyclic surfaces as holomorphic curves

Variations along the Fuchsian locus

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.Comment: 58 pages, 1 figur

The probabilistic nature of McShane's identity: planar tree coding of simple loops

In this article, we discuss a probabilistic interpretation of McShane's identity as describing a finite measure on the space of embedded paths though a point.Comment: 25 page

Ghost polygons, Poisson bracket and convexity

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 Teichm\"uller 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.Comment: 65 pages, 7 figure

Simple root flows for Hitchin representations

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.Comment: Dedicated to Bill Goldman on the occasion of his 60th birthda

Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces

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.Comment: 85 pages, 3 figures, in the version the statement of the compactness theorem 6.1 has been made more explicit for further use in some other articl