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Register a new userVariation of extremal length functions on Teichmuller space
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Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $mathbb{R}$-trees, we study the second variation of extremal length functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.
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Given a surface of infinite topological type, there are several Teichmuller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmuller space and the length-spectrum Teichmuller space. We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichmuller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmuller space is path-connected.
We prove that the every quasi-isometry of Teichmuller space equipped with the Teichmuller metric is a bounded distance from an isometry of Teichmuller space. That is, Teichmuller space is quasi-isometrically rigid.
We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $epsilon_0$-relative part of Teichmuller spaces of surfaces with boundary.
Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R^n is locally near a standard model of a flat in X. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.
The arc metric is an asymmetric metric on the Teichm{u}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurstons compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two compactifications.
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