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Convex \(\mathbb{RP}^2\) structures and cubic differentials under neck separation. (English) Zbl 1451.30084

For a closed oriented surface \(S\) of genus at least two, Labourie and the author have independently used the theory of hyperbolic affine spheres to find a canonical correspondence between convex \(\mathbb{RP}^2\) structures on \(S\) and pairs (\(\Sigma\);\(U\)) consisting of a conformal structure \(\Sigma\) on \(S\) and a holomorphic cubic differential \(U\) over \(\Sigma\).
Some geometric limits of convex \(\mathbb{RP}^2\) structures on \(S\) are obtained in which the \(\mathbb{RP}^2\) structure degenerates only along a set of simple, non-intersecting, nontrivial, non-homotopic loops. The resulting \(\mathbb{RP}^2\) structures on the loop deleted surface are classified and called regular convex \(\mathbb{RP}^2\) structures.

MSC:

30F10 Compact Riemann surfaces and uniformization
30F30 Differentials on Riemann surfaces
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