Extremal length geometry of teichmüller space

TLDR: In this paper, a point τ is a point in the Teichmuller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures.

Abstract: Assume τ is a point in the Teichmuller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures. Let be extermal lengths of two transversely realizable measured folitions on the Riemann surface R r corresponding to the point τ. There is a unique Teichmuller line along which the function is minimum. Teichmuller space embeds into projective classes of vectors of square roots of extremal lengths of simple curves on the base surface. The closure of the image of Teichmuller space under this embedding is compact. Moreover, there is a relationship between the boundary of this embedding and the boundary of the extremal length embedding properly contains the Thruston boundary.