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.
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.
We study the action of the elements of the mapping class group of a surface of finite type on the Teichmuller space of that surface equipped with Thurstons asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hype
rbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurstons classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmuller space equipped with Teichmullers metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmuller space equipped with the Weil-Petersson metric.
