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.
