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The compactification of the moduli space of convex \(\mathbb R\mathbb P^2\) surfaces. I. (English) Zbl 1085.14024

Author’s abstract: There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus \(g\) and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of W.M. Goldman [J. Differ. Geom. 31, No.3, 791–845 (1990; Zbl 0711.53033)]. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of S. Y. Cheng and S. T. Yau [Commun. Pure Appl. Math. 39, 839–866 (1986; Zbl 0623.53002)] and C.P. Wang [in: Global differential geometry and global analysis, Proc. Conf., Berlin 1990, Lect. Notes Math. 1481, 271–280 (1991; Zbl 0743.53004)] and a result of O. Dunkel [American Acad. Proc. 38, 341–370. (1902; JFM 33.0350.01)] on the asymptotics of systems of ODEs.

MSC:

14H15 Families, moduli of curves (analytic)
14P05 Real algebraic sets
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
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