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

Cross Ratios and Identities for Higher Thurston Theory

We generalise in this article the Mc Shane-Mirzakhani identities in hyperbolic geometry to arbitrary cross ratios. We give an expression of them in the case of Hitchin representations of surface groups in PSL(n, R) in a suitable choice of Fock-Goncharov coordinates.Comment: 64 pages, 5 figures The new version corrects many typos, sign errors and imprecisions. The writing has been hopefully improved. The mathematical content is identica