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Title:
Limits of cubic differentials and real projective structures under vanishing cycles on a surface
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Abstract:
If S is a closed oriented surface of genus at least 2, the space of convex real projective structures on S is equivalent to the space of pairs of conformal structures and cubic differentials on S (this is due independently to Labourie and the speaker). Thus the space of (unmarked) convex real projective structures on S is given by the total space of the vector bundle of cubic differentials over the moduli space of Riemann surfaces. We discuss an extension of this theory to the bundle of regular cubic differentials over the Deligne-Mumford compactification and investigate the corresponding regular convex real projective structures on non-compact surfaces. We will discuss these examples in terms of Higgs bundles (using work of Labourie and Baraglia) and investigate in this case the degeneration of solutions to the Hitchin equations as a sequence of cubic differentials on Riemann surfaces limit to a regular cubic differential on a nodal Riemann surface.
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