Limit points of lines of minima in Thurston’s boundary of Teichmüller space. (English) Zbl 1066.32020

Díaz, Raquel; Series, Caroline

Algebr. Geom. Topol. 3, 207-234 (2003).

Let \(S\) be a hyperbolic surface and let \(\mu\) and \(\nu\) be two measured geodesic laminations that fill up \(S\). We denote by \(l_{\mu}\) and \(l_{\nu}\) the geodesic length functions defined by \(\mu\) and \(\nu\) on the Teichmüller space \(\mathcal T\) of \(S\). S. P. Kerckhoff observed in [Duke Math. J. 65, No. 2, 187–213 (1992; Zbl 0771.30043)] that for any \(s\in(0,1)\) the function \((1-s)l_{\mu}+ sl_{\nu}\) has a unique global minimum \(m_s\) on \(\mathcal T\), and he defined the line of minima \({\mathcal L}_{\mu,\nu}\) to be the family of these minima. He stated the following fact: choosing the support of \(\mu\) to be a union of at least two disjoint closed geodesics, one can obtain an example of a line of minima \({\mathcal L}_{\mu,\nu}\) which does not converge to the projective equivalence class of \(\mu\) in Thurston’s compactification \({\mathcal PMF}\) of \(\mathcal T\).
In this paper, the authors give a proof of that statement. They show taking line of minima \({\mathcal L}_{\mu,\nu}\) with \(\mu=\sum_ {i=1}^n a_i\alpha_i\), where \(\alpha_i\) is a simple closed geodesic and \(a_i>0\) for all \(i=1,\ldots, n\), then the point \(m_s\) converges as \(s\to 0\) to the projective class of the lamination \(\sum \alpha_i\), and not to that of \(\sum a_i\alpha_i\). They also prove that if \(\mu\) is uniquely ergodic, then \(\lim_{s\to 0} m_s=[\mu]\in{\mathcal PMF}\). The results parallel results of Howard Masur, obtained in his paper [Duke Math. J. 49, 183–190 (1982; Zbl 0508.30039)], in which he proves a similar behaviour for Teichmüller geodesics.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60	Teichmüller theory for Riemann surfaces

Keywords:

Teichmüller space; line of minima; Thurson boundary; measured geodesic lamination

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