Flat Projective Structures on Surfaces and Cubic Holomorphic Differentials

By F. Labourie. 2006

144 citations recorded by the source corpus.

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a minimal surface in a symmetric space.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/acac3ce6eb52923c619ffbaf03d30136118ea222

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Polynomial cubic differentials and convex polygons in the projective plane (2014, methodology): “Labourie [Lab07] and Loftin [Lof01] have independently shown that the moduli space of convex RP(2) structures on a compact surface S of genus g > 2 can be identified with the vector bundle C(S) of holomorphic cubic differentials over the Teichmüller space T(S).”
Convex Foliated Projective Structures and the Hitchin Component for PSL(4,R) (2007, methodology): “The interpretation of the Hitchin component for PSL3(R) as holonomy representations of convex real projective structures plays an important role in independent work of F. Labourie [ 18 ] and J. Loftin [20] who associate to a convex real projective structure on � a complex structure and a cubic diffe”
Around groups in Hilbert Geometry (2013, background): “9 (Labourie [54] or Loftin [59] (compact case), Benoist-Hulin (finite volume case [13])).”
Cubic differentials and finite volume convex projective surfaces (2013, background): “The aim of this section, which basically follows the presentation by Labourie in [10], is to define the Pick form U of an affine 2–dimensional sphere (Remark-Definition 4.”
Survey on Affine Spheres (2008, methodology): “Under this map Labourie [39] has identified Hitchin’s cubic differential with (a constant multiple of) C.P. Wang’s cubic differential U coming from the hyperbolic affine sphere: The RP2 structure coming from the affine sphere determined by (Σ,− 1 12 U) has holonomy determined by Hitchin’s Higgs bund”
Quiver algebras as Fukaya categories (2013, background): “At least for An-fibred threefolds, it seems natural following [33] to expect the “higher Teichmüller space”, i.”
An invitation to higher Teichm\"uller theory (2018, methodology): “For Hitchin components Labourie made a very precise conjecture, based on Hitchin’s parametrization of the Hitchin component.”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, unknown): “Labourie’s conjecture was proven independently by Loftin [Lof01] and Labourie [Lab07] for G = SL(3,R), and then recently by Labourie [Lab17] for other split real Lie groups of rank 2 (namely, PSp(4,R) and G2).”
Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, background): “The proof for PSL(3,R) is independently due to Labourie [70] and Loftin [74].”