continuity of length function $l: T(X) \times MF \to \mathbb R$

Question

Let $T(X)$ be the Teichmuller space of a closed Riemann surface $X$ of genus $g \geq 2,$ and $MF$ the space of equivalence classes of measured foliations. Then we have a length function $l: T(X) \times MF \to \mathbb R,$ $l_\sigma (\mathcal F).$ The question is, is there a basic and convenient way to see the continuity of this function $l$. Here $T(X)$ is given the usual topology and $MF$ the topology of intersection numbers.

Analogously, let $Q (X)$ be the space of holomorphic quadratic differentials on $X.$ Each $q \in Q (X)$ gives a metric $|q|$ on $X,$ and we also have a length function $L: Q(X) \times MF \to \mathbb R,$ $l_q (\mathcal F).$ A same question is, is there a basic and convenient way to see the continuity of this function $L$. Here $Q(X)$ is given the topology induced by the $L^1$-norm and $MF$ is given the intersection number topology.

Answer 1

I suppose that to "see" these continuities, you are not asking about a proof, but about some intuition.

For the first question, I like the intuition one gets from Bonahon's paper MR0931208, under which $MF$ and $T(X)$ are both embedded in a common space on which an obviously continuous intersection number is defined. This is the space of $\pi_1(X)$-invariant Borel measures defined on the "Mobius band beyond infinity" $M = ((\partial \pi_1(X) \times \partial \pi_1(x)) - \Delta) / (\mathbb{Z}/2)$, using the weak topology on the space of Borel measures on $M$. $T(X)$ embeds via the Liouville measure of a hyperbolic metric, and $MF$ embeds in an obvious fashion. Given two such measures $\mu,\nu$, using a local Fubini product description Bonahon defines a product measure on $M \times M$, mods out by $\pi_1(M)$, and the total mass of the quotient; the result is "obviously" continuous, by simple facts about weak topologies on measure spaces.

Another way to say almost the same thing is to think of $T(X)$ as a continuously varying family of hyperbolic structures, and of $MF$ as a continuously varying family of measured geodesic laminations, then take the local Fubini product of the transverse measure on the lamination and the length measure along leaves from the hyperbolic structure, integrate, and you get the intersecton number; the whole picture varies continuous in both the $T(X)$ variable and the $MF$ variable, the Fubini product measure varies continuously in the weak topology on Borel measures, and its total measure varies continuously.

For the second question, one can see continuity in a similar manner. Think of $Q(X)$ as a continuously varying family of singular Euclidean structures on $X$. For elements of $MF$, think of straightening them in each singular Euclidean structure to become "singular Euclidean measured geodesic laminations". Convince yourself that this whole picture varies continuously. Again, take Fubini product measures locally, integrate, and you get the intersection number. The one twist here is that when you straighten an element of $MF$ in a singular Euclidean structure, leaves need not stay disjoint, but that's ok because all you really need to do is to pull back the Euclidean length measure along leaves to some abstract model of the lamination.

Answer 2

Here's another answer that may be more satisfying: Use convexity. Short summary: length of weighted simple closed multi-curves defines a convex function on the space of measured foliations with respect to any system of natural coordinates. A convex function defined at rational points automatically has a unique continuous extension to real values of the parameters, which in this case is all measured foliations.

I'm happy to give more details, but I'd like some confirmation that someone is reading this first...