## 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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