I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me out.

First let me introduce it: in this (unpublished) article Thurston, between other things, gave two metrics on Teichmüller space of surfaces of genus $g$:

$K(g,h):=\sup_{\alpha\in \mathbb{S}}log(\frac{l_g(\alpha)}{l_h(\alpha)})=\sup_{\alpha\in PML(S)}log(\frac{l_g(\alpha)}{l_h(\alpha)})$ where $\mathbb{S}$ is the set of isotopy classes of simple curves on $S$ which is dense in $PML(S)$, the set of projective classes of measured laminations on $S$.

$L(g,h):=\inf_{Diff_0^+(S)}log(L(\phi))$ where $L(\phi)$ is the Lipschitz constant of $\phi$ with respect to hyperbolic metrics $g$ and $h$.

Thurston proved the equality (theorem 8.5) $K(g,h)=L(g,h)$ for every couple of hyperbolic metrics $g$ and $h$.

Theorem 8.1 is the first step in order to obtain this equality: it states that if $\lambda\in ML(S)$ is such that the ratio $\frac{l_g(\lambda)}{l_h(\lambda)}$ is maximal in a neighborhood of $\lambda$ in $ML(S)$ then there exists a map isotopic to the identity from a neighborhood of the geodesic realization of $\lambda$ in $g$ to the geodesic realization of $\lambda$ in $h$ with Lipschitz constant $\frac{l_g(\lambda)}{l_h(\lambda)}$.

My unclear points regard proof of theorem 8.1:

1) In Proposition 6.4 Thurston computes the tangent space of $ML(S)$ (as a piece-wise linear space, via the structure given by train tracks coordinates) at $\lambda$ : it is the union of the cones $V(\mu)$ where $\mu$ is a chain recurrent lamination containing $\lambda$ and such that $cut(\mu)$ admits a transverse measure of full support.

My question is: are there other references for this construction? Thurston's proofs are not so detailed (expecially about the function $cut$). I've read Harer's book about train tracks but he doesn't mention the tangent space of $ML(S)$. I also know Bonahon studied the tangent space to $ML(S)$ but only in terms of laminations with transverse Holder structure.

2) In theorem 7.1 Thurston computes the first order approximation to the length function $l_g$ on $ML(S)$: for every lamination $\lambda'\in V(\mu)\subset T_\lambda(ML(S))$ the first order approximation to $l_g(\lambda')$ is $i(\lambda',F_g(\mu))$ ($F_g(\mu)$ is the horocyclic foliation of $\mu$ with respect to the metric $g$). I've not followed every detail in the proof, but the result is intuitively clear.

What is not clear to me is how Thurston uses this in the proof of Theorem 8.1: if $\lambda$ is maximal for $\frac{l_g}{l_h}$ then the derivative of $\frac{l_g}{l_h}$ is zero in $\lambda$. But then Thurston says that, "by theorem 7.1, it implies that the measure class of $F_g(\lambda)$ is the multiple $\frac{l_g(\lambda)}{l_h(\lambda)}$ of the measure class of $F_h(\lambda)$" My question is: How does it follow from $d_\lambda(\frac{l_g}{l_h})=0$ that $F_g(\lambda)=\frac{l_g(\lambda)}{l_h(\lambda)}F_h(\lambda)$?


1 Answer 1


Here is an answer to question (2). Let $\lambda' \in V(\lambda)$ be a lamination in the near subspace of the tangent cone at $\lambda$. Let $\alpha(t)$ have $\alpha(0) = \lambda$ and $\alpha'(0) =\lambda'$. As $l_h(\lambda)$ is nonzero, the fact that $\frac{d}{dt} l_g(\alpha(t)) / l_h(\alpha(t)) = 0$ implies with the quotient rule and Theorem 7.1 that $$0 = l_h(\lambda) (D l_g)_\lambda(\lambda') - l_g(\lambda)(Dl_h)_\lambda(\lambda') = l_h(\lambda) i(\lambda', F_g(\lambda)) - l_g(\lambda)i(\lambda',F_h(\lambda)).$$ So for all $\lambda'$ in the near subspace of the tangent cone to $ML$ at $\lambda$ we have $i(\lambda', F_g(\lambda)) = \frac{l_g(\lambda)}{l_h(\lambda)} i(\lambda', F_h(\lambda)).$ This determines the measure class of $F_g(\lambda)$ among foliations supported in a neighborhood of stable topological type containing $\lambda$.


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