增广Teichmuller空间与Orbifolds

增广Teichmüller空间上的Thurston拉回映射及其应用

Thurston's pullback map on the augmented Teichmüller space and applications

设$f$是二维拓扑球面的后临界有限分支自覆盖.这样的映射导出有限维Teichmüller空间的解析自映射$σ_f$。我们证明了这个映射连续扩张到增广Teichmüller空间，并给出了这个扩张的一个明确的构造。这使我们能够表征瑟斯顿的拉回映射附近的不变地层的边界的增广Teichmüller空间的动力学。不变的边界地层的分类是用来证明一个猜想朝圣者和推断进一步的性质瑟斯顿的拉回地图。我们的方法也产生了新的证明瑟斯顿定理和朝圣者的典型障碍定理。

On the symplectic structure over a moduli space of orbifold projective structures

Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent bundle of the Teichmüller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification is proved to be compatible with these symplectic structures. Similar results are …

Universal commensurability augmented Teichmüller space and moduli space

It is known that every finitely unbranched covering $α:\widetilde{S}_{g(α)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $Γ_α$ from the Teichmüller space $T(S)$ to the Teichüller space $T(\widetilde{S}_{g(α)})$. Actually, it has been showed that the isometric embedding $Γ_α$ can be extended isometrically to the augmented Teichmüller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichmüller spaces, where the index runs over all finitely unbranched coverings of $S$. Th…

On Kawai theorem for orbifold Riemann surfaces

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic from on $\mathrm{PSL}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric…

Teichmüller曲线上的曲率

Curvatures on the Teichmüller curve

Teichmüller曲线是闭黎曼曲面的Teichmüller空间上的纤维空间，其中Teichmüller空间中一点上的纤维是基础曲面。我们推导出Teichmüller曲线的截面曲率公式。特别地，我们的方法可以应用于研究Weil-Petersson测地线作为三流形的几何，以及沿着Weil-Petersson测地线的增广Teichmüller空间的无穷远曲率的退化，以及在这个三流形中的双曲曲面的极小性。

Universal degeneration of Riemann surfaces and fibered complex surfaces

Based on Teichmüller theory, we construct a degenerating family $\overline{Y}_g^{orb} \rightarrow \overline{M}_g^{orb}$ over the Deligne-Mumford compactification of the moduli space with the natural orbifold structure such that any fibered complex surface admitting unstable fibers can be pulled back from this family.