Bulge Derivatives and Deformations of Convex Real Projective Structures on Surfaces

Title of dissertation: TWIST-BULGE DERIVATIVES AND DEFORMATIONS OF CONVEX REAL PROJECTIVE STRUCTURES ON SURFACES Terence Dyer Long, Doctor of Philosophy, 2015 Dissertation directed by: Professor Scott Wolpert Department of Mathematics Let S be a closed orientable surface with genus g > 1 equipped with a convex RP structure. A basic example of such a convex RP structure on a surface S is the one associated to a hyperbolic structure on S, and in this special case Wolpert proved formulas for computing the Lie derivatives tαlβ and tγtαlβ, where tα is the Fenchel-Nielsen twist vector field associated to the twist along a geodesic α, and l∗ is the hyperbolic geodesic length function. In this dissertation, we extend Wolpert’s calculation of tαlβ and tγtαlβ in the hyperbolic setting to the case of convex real projective surfaces; in particular, our tα is the twist-bulge vector field along geodesic α coming from the parametrization of the deformation space of convex RP structures on a surface due to Goldman, and our geodesic length function l∗ is in terms of a generalized cross-ratio in the sense of Labourie. To this end, we use results due to Labourie and Fock-Goncharov on the existence of an equivariant flag curve associated to Hitchin representations, of which convex real projective surfaces are an example. This flag curve allows us to extend the notions arising in the hyperbolic case to that of convex real projective structures and to complete our generalization of Wolpert’s formulas. TWIST-BULGE DERIVATIVES AND DEFORMATIONS OF CONVEX REAL PROJECTIVE STRUCTURES ON SURFACES by Terence Long Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Advisory Committee: Professor Scott A. Wolpert, Chair Professor John J. Millson Professor William M. Goldman Professor Karin H. Melnick Professor Thomas D. Cohen To Islem, my patient and long-suffering wife,