Complex projective structures with Schottky holonomy

A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from pi_1(S) to the Schottky group. We will show that an arbitrary projective structure with the same holonomy representation is obtained by (2 pi-)grafting the basic structure described above.Comment: 52 pages, 14 figure

Neck-Pinching of CP1CP^1-structures in the PSL(2,C)PSL(2,C)-character variety

Let S be a closed oriented surface of genus at least two. We consider a path of $CP^1$-structures $C_t$ on S leaving every compact subset in the deformation space of (marked) $CP^1$-structures on S, such that its holonomy converges in the PSL(2, C)-character variety. In this setting, it is known that the complex structure $X_t$ of $C_t$ also leaves every compact subset in the Teichm\"uller space. In this paper, under the assumption that $X_t$ is pinched along a single loop m, we describe the limit of $C_t$ in terms of the developing maps, holomorphic quadratic differentials, and pleated surfaces. Moreover, we give an example of such a path $C_t$ whose the limit holonomy is the trivial representation in the character variety.Comment: 52 pages, 22 figure

Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties

We consider the space of ordered pairs of distinct ℂP¹ -structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal. In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (SL₂ ℂ-opers) is a non-empty discrete set, which is closely related to the mapping.The version of record of this article, first published in Geometric and Functional Analysis, is available online at Publisher’s website: https://doi.org/10.1007/s00039-023-00653-

Realisation of bending measured laminations by Kleinian surface groups

For geometrically finite Kleinian surface groups, Bonahon and Otal proved the existence part, and partly the uniqueness part of the bending lamination conjecture. In this paper, we generalise the existence part to general Kleinian surface groups including geometrically infinite ones. We furthermore prove the compactness of the set of Kleinian surface groups realising an arbitrarily fixed data of bending laminations and ending laminations. Our proof is independent of that of Bonahon and Otal.Comment: 26 page

2 \pi-grafting and complex projective structures, I

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2\pi-graftings produce all projective structures on $S$ with arbitrarily fixed holonomy (Grafting Conjecture). In this paper, we show that the conjecture holds true "locally" in the space $GL$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $GL$ in the Thurston coordinates. In the sequel paper, using this local solution, we prove the conjecture for generic holonomy.Comment: 57 pages, 10 figures. To appear in Geometry & Topolog