Affine spheres and convex RPn-manifolds

By John C. Loftin. 2001

139 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2357e1b60ac6511179d77da7e241b54cf92b665a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Convex Foliated Projective Structures and the Hitchin Component for PSL(4,R) (2007, background): “Loftin [20] who associate to a convex real projective structure on Σ a complex structure and a cubic differential on Σ.”
Locally Homogeneous Geometric Manifolds (2010, methodology): “Using analytic techniques, Labourie [122] and Loftin [126], independently, described C(Σ) as a cell in a quite different way.”
THE DEFORMATION SPACES OF CONVEX RP²-STRUCTURES ON 2-ORBIFOLDS (2001, methodology): “This can be seen by a Klein model of hyperbolic geometry; a hyperbolic structure naturally induces a canonical convex RP2-structure on the orbifold.”
An invitation to higher Teichm\"uller theory (2018, methodology): “This conjecture has been proven by Labourie [77] and Loftin [88] for G = PSL(3,R) and by Labourie [79], using Higgs bundle methods, for all split real Lie groups of rank 2.”
Cyclic surfaces and Hitchin components in rank 2 (2014, methodology): “For SL(3,R), this corollary was obtained by Loftin in [33] and the author in [30] (announced in [28]) Theorem 1.”
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).”
Survey on Affine Spheres (2008, background): “We give two constructions in [48, 49]: For any cubic differential U with three poles of order ≤ 3 on CP1, a version of Ţiţeica’s equation (for an appropriate background metric) ∆η + 4‖U‖2e−2η − 2eη − 2κ = 0 can be solved to produce a hyperbolic affine sphere structure on CP1 minus the pole set…”
Bernstein theorem and regularity for a class of monge-ampére equations (2013, background): “In early 2009, John Loftin told us that the transform coincides with the conormal map in affine geometry, which transforms a hyperbolic affine sphere to its dual sphere [23]; see Example 4 below.”
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].”