arXiv subjects
Publications and source records attributed to Shinpei Baba.
Let $\Sigma$ be a closed orientable surface of genus at least two, and let $X, Y$ be distinct marked Riemann surface structures on $\Sigma$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably infinite pairs of ${\rm CP}^1$-structures on $X$ and on $Y$ sharing holonomy $\pi_1(\Sigma) \to {\rm PSL}_2 {\rm C}$.
We consider the mapping $b_L\colon\mathcal{T} \to \chi$ from the Fricke-Teichm\"uller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $\chi$ of the surface, obtained by bending Fuchsian representations along a fixed measured lamination $L$. We prove that this mapping is an equivariant symplectic real-analytic embedding, and, for almost all measured laminations, proper. We also show that this ``bending map'' $b_L\colon \mathcal{T} \to \chi$ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of $\mathcal{T}$ into the Morgan-Shalen boundary of $\chi$. Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety ${\rm Im} b_L$ into the product variety $\chi \times \chi$ by the diagonal mapping twisted by complex conjugation. Then we construct a closed $\mathbb{C}$-symplectic complex-analytic subvariety of $\chi \times \chi$ containing $\mathrm{Im} b_L$ as a half-dimensional real-analytic subvariety.
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.
We consider the space of ordered pairs of distinct $\mathbb{C}P^1$-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 Teichm\"uller 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 Poincar\'e holonomy varieties (${\rm SL}_2\mathbb{C}$-opers) is a non-empty discrete set, which is closely related to the mapping.
We characterize a certain neck-pinching degeneration of (marked) $CP^1$- structures on a closed oriented surface S of genus at least two. Namely, we consider a path $C_t$ of $CP^1$-structures 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 limit holonomy is the trivial representation in the character variety.
Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures, based on Kamishima-Tan and Kulkani-Pinkall. We, in addition, give independent proofs for the following well-known theorems on $\mathbb{C}{\rm P}^1$-structures by means of pleated surfaces given by the parameterization. (1) Goldman's Theorem on $\mathbb{C}{\rm P}^1$-structures with quasi-Fuchsian holonomy. (2) The path lifting property of developing maps in their domains of discontinuity in $\mathbb{C}{\rm P}^1$.
Let S be an oriented closed surface of genus at least two. We show that, given a generic representation in the PSL(2,C)-character variety of S, (2π-)graftings produce all projective structures on S with the holonomy representation.
Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2π-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.
Let $S$ be a closed oriented surface of genus $g\geq 2$. Fix an arbitrary non-elementary representation $ρ\colonπ_1(S)\to {\rm SL}_2(\mathbb{C})$ and consider all marked (complex) projective structures on $S$ with holonomy $ρ$. We show that their underlying conformal structures are dense in the moduli space of $S$.
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.
Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on (S, C), along which a grafting can be done.