We review several applications of Douady-Earle section to holo- morphic motions over infinite dimensional parameter spaces. Using Douady- Earle section we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomor- phic motions in complex analysis.

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Douady-Earle section, holomorphic motions, and some applications

It is known that every finitely unbranched holomorphic covering π:$$\pi :\tilde{S} \to S$$ of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding $${{\rm{\Phi}}_{\rm{\pi}}}:Teich\left(S \right) \to Teich\left({\widetilde{S}} \right)$$. By the mutual relations between Strebel rays in Teich(S) and their embeddings in $$Teich\left({\widetilde{S}} \right)$$, we show that the 1st-strata space of the augmented Teichmuller space $$\widehat{Teich}\left(S \right)$$ can be embedded in the augmented Teichmuller space $$\widehat{Teich}\left({\widetilde{S}} \right)$$ isometrically. Furthermore, we show that Φπ induces an isometric embedding from the set Teich(S) B (∞) consisting of Busemann points in the horofunction boundary of Teich(S) into $$Teich{\left({\widetilde{S}} \right)_B}\left(\infty \right)$$ with the detour metric.

Isometric Embeddings of Subsets of Boundaries of Teichmüller Spaces of Compact Hyperbolic Riemann Surfaces

Any Beltrami coefficient $\mu $ on R represents a point $[\mu ]_T$ in Teichmuller space T(R) of a hyperbolic Riemann surface R and a point $[\mu ]_B$ in the tangent space of T(R) at the base point. The problem is whether $\mu $ represents a Strebel point is equivalent to $\mu $ represents an infinitesimal Strebel point. We prove that for any $\mu $, there exists an element $\nu $, such that $[\nu ]_T=[\mu ]_T$ and $[\nu ]_B$ is an infinitesimal Strebel point, when R is the unit disk.

On the Relation Between Strebel Points and Infinitesimal Strebel Points

It is known that every finitely unbranched covering $\alpha:\widetilde{S}_{g(\alpha)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Gamma_{\alpha}$ from the Teichmuller space $T(S)$ to the Teichuller space $T(\widetilde{S}_{g(\alpha)})$. Actually, it has been showed that the isometric embedding $\Gamma_{\alpha}$ can be extended isometrically to the augmented Teichmuller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichmuller spaces, where the index runs over all finitely unbranched coverings of $S$. Then, we show that the action of the universal commensurability modular group $Mod_{\infty}(S)$ can extend isometrically on $\widehat{T}_{\infty}(S)$. Furthermore, for any $X_{\infty}\in T_{\infty}(S)$, its orbit of the action of the universal commensurability modular group $Mod_{\infty}(S)$ on the universal commensurability augmented Teichmuller space $\widehat{T}_{\infty}(S)$ is dense. Finally, we also construct a directed limit $\widehat{M}_{\infty}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut(\pi_{1}(S))$ of $Mod_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to produce $\widehat{M}_{\infty}(S)$ as the quotient.

Universal commensurability augmented Teichm\"uller space and moduli space

Higher Bers maps and BMO-Teichmüller space

We prove, by using the main inequality of Reich and Strebel, that any n K-quasiconformal germs defined on n disjoint domains in the Riemann sphere can be glued by one (K + e)-quasiconformal homeomorphism, where e is a positive number which can go to zero as the domains of germs shrinking to n points. This generalizes a result in [8] where only the case K = 1 has been considered.

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A gluing theorem for quasiconformal mappings

It is known that every finitely unbranched holomorphic covering $\pi:\widetilde{S}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Phi_{\pi} :Teich(S)\rightarrow Teich(\widetilde{S})$. By the mutual relations between Strebel rays in $Teich(S)$ and their embeddings in $Teich(\widetilde{S})$, we show that the augmented Teichmuller space $\widehat{Teich}(S)$ can be isometrically embedded in the augmented Teichmuller space $\widehat{Teich}(\widetilde{S})$.

The isometric embedding of the augmented Teichmüller space of a Riemann surface into the augmented Teichmüller space of its covering surface

It is shown that the complex dilatation of the Douady-Earle extension of a strongly symmetric homeomorphism induces a vanishing Carleson measure on the unit disk D. As application, it is proved that the VMO-Teichmuller space is a subgroup of the universal Teichmuller space.

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Yi Qi

Papers

A gluing theorem for quasiconformal mappings

The isometric embedding of the augmented Teichmüller space of a Riemann surface into the augmented Teichmüller space of its covering surface

Douady-Earle extension of the strongly symmetric homeomorphism