Let $S$ be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural 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 $\Sigma$. We consider geometric limits of convex $\mathbb{RP}^2$ structures on $S$ in which the $\mathbb{RP}^2$ structure degenerates only along a set of simple, non-intersecting, nontrivial, non-homotopic loops $c$. We classify the resulting $\mathbb{RP}^2$ structures on $S - c$ and call them regular convex $\mathbb{RP}^2$ structures. Under a natural topology on the moduli space of all regular convex $\mathbb{RP}^2$ structures on $S$, this space is homeomorphic to the total space of the vector bundle over $\overline{M}_g$ each of whose fibers over a noded Riemann surface is the space of regular cubic differentials. The proof relies on previous techniques of the author, Benoist–Hulin, and Dumas–Wolf, as well as some details due to Wolpert of the geometry of hyperbolic metrics on conformal surfaces in $\overline{M}_g$.

中文翻译：

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凸分离下的凸\ mathbb {RP} ^ 2 $结构和三次微分

令$ S $是至少两个属的闭合定向曲面。Labourie和作者独立地使用了双曲仿射球理论在$ S $上的凸$ \ mathbb {RP} ^ 2 $结构与由保形结构$组成的对$（\ Sigma，U）$之间找到了自然对应关系。 \ S $上的\ Sigma $和$ \ Sigma $上的全纯三次微分$ U $。我们考虑$ S $上凸$ \ mathbb {RP} ^ 2 $结构的几何极限，其中$ \ mathbb {RP} ^ 2 $结构仅沿着一组简单，不相交，不平凡，非同构的退化循环$ c $。我们对$ S-c $上的结果$ \ mathbb {RP} ^ 2 $结构进行分类，并将它们称为规则凸形$ \ mathbb {RP} ^ 2 $结构。在所有自然凸拓扑$ \ mathbb {RP} ^ 2 $结构在$ S $的模空间上的自然拓扑下，这个空间是等价于向量\ overline {M} _g $上向量束的总空间的，每个向量束在节点Riemann曲面上的纤维都是规则三次微分的空间。该证明依赖于作者Benoist–Hulin和Dumas–Wolf的先前技术，以及由于$ \ overline {M} _g $中的共形表面上的双曲线度量的几何形状的Wolpert所引起的一些细节。