The Compactification of the Moduli Space of Convex ℝℙ2 Surfaces, I

Loftin, John

doi:10.4310/jdg/1115669512

Cited by 29 publications

(71 citation statements)

References 33 publications

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“…The structurally similar Bochner equation governing harmonic maps to hyperbolic surfaces had an analogous development of error estimates in [Min92], [Wol91], and [Han96]. Crucial to refined error estimates are the use of sub-and supersolutions, first constructed in the present setting by Loftin [Lof04]; for the analogous Bochner equation and for open Riemann surfaces, this technique began with [Wan92] (see also [WA94]). …”

Section: Introductionmentioning

confidence: 99%

“…the affine structure equations restricted to a curve). The technique we use here was introduced by Loftin in [Lof04]; there was no lower-dimensional Bochner equation analogue to this technique.…”

Section: Introductionmentioning

confidence: 99%

“…However, we build upon a substantial history of error estimates for Wang's equation and other geometric PDE. While both Loftin [Lof01] and Labourie [Lab07] focused on the Wang equation in their studies of affine spheres, Loftin ([Lof04], [Lof07], [Lof15]) began and developed a theory of error estimates for this equation. The structurally similar Bochner equation governing harmonic maps to hyperbolic surfaces had an analogous development of error estimates in [Min92], [Wol91], and [Han96].…”

Section: Introductionmentioning

confidence: 99%

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Polynomial cubic differentials and convex polygons in the projective plane

2015

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Abstract. We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d + 3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP 2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmüller space.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polynomial cubic differentials and convex polygons in the projective plane

2015

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“…He gave a condition in terms of the affine metric for a two-dimensional surface to be an affine sphere that involves conformal geometry. See also Labourie [33] and Loftin [37].…”

Section: Hitchin's Parametrizationmentioning

confidence: 99%

Convex real projective structures and Hilbert metrics

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Papadopoulos

2014

IRMA Lectures in Mathematics and Theoretical Physics

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Abstract. We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in SL(3, R). In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.

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“…In particular, we use Simon-Wang's developing map and techniques of ODEs to calculate the holonomy and other natural invariants of affine flat structure. (This basic plan of first solving for a conformal factor and then applying a developing map and ODE techniques to characterize the relevant geometric structure near a singular point on a surface was first carried out in [25], where asymptotics for singular convex real projective structures were investigated using hyperbolic affine spheres.) Finally we use Blaschke's holomorphic characterization to find precise asymptotics of the metric and the affine flat structure.…”

Section: Theorem 1 Given Any Holomorphic Cubic Differential U On Cpmentioning

confidence: 99%