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Register a new userTeichmuller spaces of generalized symmetric homeomorphisms
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We introduce the concept of a new kind of symmetric homeomorphisms on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.
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In a paper of Cui and Zinsmeister the equivalence among three definitions of BMO-Teichmuller spaces associated with a Fuchsian group was proven using the Douady-Earle extension operator. In this paper, we show that these equivalences are actually biholomorphisms. It was further shown in the above quoted paper that the Douady-Earle extension operator is continuous at the origin. We improve this result by showing G^ateaux-differentiability at this point.
Let $Omega$ be an internal chord-arc Jordan domain and $varphi:mathbb SrightarrowpartialOmega$ be a homeomorphism. We show that $varphi$ has finite dyadic energy if and only if $varphi$ has a diffeomorphic extension $h: mathbb Drightarrow Omega$ which has finite energy.
Let $Omega subset mbr^2$ be an internal chord-arc domain and $varphi : mbs^1 rightarrow partial Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h : mbd rightarrow Omega$ of $varphi .$ We study the relationship between weighted integrability of the derivatives of $h$ and double integrals of $varphi$ and of $varphi^{-1} .$
We construct an Ahlfors-Bers complex analytic model for the Teichmuller space of the universal hyperbolic lamination (also known as Sullivans Teichmuller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivans Teichmuller space is Kahler isometric biholomorphic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichmuller space of this surface; i.e. We find natural Kahler coordinates for the Sullivans Teichmuller space. This is the main result. As a corollary, we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivans Teichmuller space contained in the universal one.
We prove that the Teichmuller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmuller metric) is almost isometric to the Teichmuller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmuller metric).
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