Lines of minima with no end in Thurston’s boundary of Teichmüller space

Iguchi, Yuki

doi:10.1090/S1088-4173-2012-00240-8

Lines of minima with no end in Thurston’s boundary of Teichmüller space
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by Yuki Iguchi

Conform. Geom. Dyn. 16 (2012), 22-43

DOI: https://doi.org/10.1090/S1088-4173-2012-00240-8

Published electronically: March 7, 2012

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Abstract:

Let $\nu ^+$ and $\nu ^-$ be two measured laminations which fill up a hyperbolic surface. Kerckhoff [Duke Math. J. 65 (1992), 187–213] defines a line of minima as a family of surfaces where convex combinations of the hyperbolic length functions of $\nu ^+$ and $\nu ^-$ are minimum. This is a proper curve in the Teichmüller space. We show that there exists a line of minima which does not converge in the Thurston compactification of the Teichmüller space of a compact Riemann surface. We also show that the limit set of the line of minima is contained in a simplex on the Thurston boundary.

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