Abstract: I will present results on the first order behavior of certain geometric quantities at the Fuchsian locus of the Hitchin component. These may be regarded as "higher" analogs of classical formulas in Teichmueller theory. A precise description in terms of holomorphic differentials of the tangent spaces at the Fuchsian locus to the moduli of Higgs bundles, Opers, and the Hitchin component. leads to a generalization of Ahlfors' result on the vanishing of the first order variation of the harmonic metric for certain good variations. As an application, I will explain the relationship between Poincare series and the hamiltonian vector fields associated to holonomy of Hitchin representations about simple closed curves. Finally, I will state a precise relationship between the Pressure metric and the Petersson pairing along the Fuchsian locus.
This is joint work with Francois Labourie.