Flat metrics, cubic differentials and limits of projective holonomies

F. Labourie and the author independently have shown that a convex real projective structure on an oriented closed surface S of genus at least two is equivalent to a pair of a conformal structure plus a holomorphic cubic differential. Along certain paths, we find the limiting holonomy of convex real projective structures on a surface S corresponding corresponding to a given fixed conformal structure S and a holomorphic cubic differential λ U0 as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda\to\infty$$\end{document} . We explicitly give part of the data needed to identify the boundary point in Inkang Kim’s compactification of the deformation space of convex real projective structures. The proof follows similar analysis to that studied by Mike Wolf is his application of harmonic map theory to reproduce Thurston’s boundary of Teichmüller space.