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Register a new userTeichmuller theory of the universal hyperbolic lamination
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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.
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We introduce a function model for the Teichmuller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmuller space. We prove that the identity map from the Teichmuller space equipped with the usual Teichmuller metric to the Teichmuller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichmuller space equipped with this new metric to the Teichmuller space equipped with the usual Teichmuller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichmuller metric on the Teichmuller space. We give a remark about the pressure metric and the Weil-Petersson metric.
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.
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.
Teichmullers problem from 1944 is this: Given $xin [0,1)$ find and describe the extremal quasiconformal map $f:IDtoID$, $f|partial ID=identity$ and $f(0)=-xleq 0$. We consider this problem in the setting of minimisers of $L^p$-mean distortion. The classical result is that there is an extremal map of Teichmuller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x eq 0$. For the $L^p$-norm, when $p=1$ it is known that there can be no locally quasiconformal minimiser unless $x=0$. Here we show that for $1leq p<infty$ there is a minimiser in a weak class and an associated Ahlfors-Hopf holomorphic quadratic differential with a pole of order $1$ at $f(0)=r$. However, this minimiser cannot be in $W^{1,2}_{loc}(ID)$ unless $r=0$ and $f=identity$. Hence there is no locally quasiconformal minimiser. A similar statement holds for minimsers of the exponential norm of distortion. We also use our earlier work to show that as $ptoinfty$, the weak $L^p$-minimisers converge locally uniformly in $ID$ to the extremal quasiconformal mapping, and that as $pto 1$ the weak $L^p$-minimisers converge locally uniformly in $ID$ to the identity.
We introduce $Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Theta$-positivity generalizes at the same time Lusztigs total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Theta$-positive structure. We describe key aspects of $Theta$-positivity and make a connection with representations of surface groups and higher Teichmuller theory.
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