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PDE and Differential Geometry Seminar
Real Projective Structures, Cubic Differentials, and Degenerations Under Neck Separations
John Loftin (Rutgers/Harvard)

Monday, November 4, 2019
2:30pm – 3:30pm

Storrs Campus
MONT 214

A convex real projective structure on a surface is given by writing it a quotient of a convex domain D in the real projective plane by a representation of the fundamental group into PGL(3,R) which acts properly discontinuously on D. The geometry is determined an invariant convex surface in the cone over D, a hyperbolic affine sphere H, as studied by Cheng-Yau and others. H in turn is determined by two elliptic PDEs, first by a real Monge-Ampere equation on D whose solution determines H directly, and second by a semilinear equation on S in terms of a conformal structure and holomorphic cubic differential C. This semilinear equation serves as an integrability condition for H with an appropriate initial frame. C=0 is equivalent to H being a hyperboloid and the real projective structure being Fuchsian under the Klein model of hyperbolic space.

We will use these cubic differentials to explain the moduli space of convex real projective structures as a vector bundle over the moduli space of Riemann surfaces (this is also independently due to Labourie), and also to extend this picture to the boundary of the Deligne-Mumford compactification of moduli space.


PDE and Differential Geometry Seminar (primary), UConn Master Calendar

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