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Home » Geometry and Analysis of Surface Group Representations: Cubic Differentials and Limits of Convex $RP^2$ Strucures under Neck Pinches

Seminar

Geometry and Analysis of Surface Group Representations: Cubic Differentials and Limits of Convex $RP^2$ Strucures under Neck Pinches March 04, 2015 (11:00 AM PST - 12:30 PM PST)
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Location: MSRI: Simons Auditorium
Speaker(s) John Loftin (Rutgers University)
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Labourie and I independently proved that on a closed oriented surface $S$ of genus $g$ at least 2, a convex real projective structure is equivalent to a pair $(\Sigma,U)$, where $\Sigma$ is a conformal structure and $U$ is a holomorphic cubic differential. It is then natural to allow $\Sigma$ to go to the boundary of the moduli space of Riemann surfaces.  The bundle of cubic differentials then extends over the boundary to form the bundle of regular cubic differentials, which is an orbifold vector bundle over the Deligne-Mumford compactification $\bar{\mathcal M}_g$ of moduli space.



We define regular convex real projective structures corresponding to the regular cubic differentials over nodal Riemann surfaces and define a topology on these structures. Our topology is an extension of Harvey's use of the Chabauty topology to analyze $\bar {\mathcal M}_g$ via limits of Fuchsian groups.  The main theorem is that the total space of the bundle of regular cubic differentials over $\bar {\mathcal M}_g$ is homeomorphic to the space of regular real projective structures.  The proof involves several analytic inputs: a recent result of Benoist-Hulin on the convergence of some invariant tensors on families of convex domains converging in the Gromov-Hausdorff sense, a recent uniqueness theorem of Dumas-Wolf for certain complete conformal metrics, and some old techniques of the author to specify the real projective end of a surface in terms of the residue of a regular cubic differential.

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