Does the Teichmüller space of the pair of pants admit a continuous global section?

Question

Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth isotopies).

The function $L:H(P) \to (\mathbb{R}^+)^3$ which associates to each metric the length of the three boundary curves (where the order has been fixed) descends as a homeomorphism to the quotient $T(P)$.

It is known that $L$ admits continuous local sections. That is, any point in $T(P)$ has a neighborhood on which one can choose a representative of each class in continuous manner. A proof of this is given in Theorem 3.5 of Fathi, Laudenbach, Poenaru, et al's book "Thurston's work on Surfaces".

My question: Does the Teichmüller space of the pair of pants admit a continuous global section?

Answer 1

Since the fiber is contractible, there is a global section. The details are in the papers

Earle, Clifford J.; Eells, James, A fibre bundle description of Teichmüller theory. J. Differential Geometry 3 (1969) 19–43.

Earle, C. J.; Schatz, A., Teichmüller theory for surfaces with boundary. J. Differential Geometry 4 (1970) 169–185.

which treat this question in much greater generality.

Answer 2

Yes, this is true. The proof is really the same as the hyperbolic trigonometry proof which yields existence of continuous local sections.

The first step is that given $(x,y,z) \in (\mathbb{R}^+)^3$ one can construct a right angled hyperbolic hexagon $Q(x,y,z) \subset \mathbb{H}^2$ having side lengths $x/2,x',y/2,y',z/2,z'$, so that the missing side lengths $x',y',z'$ are given by continuous (even analytic) formulas in $x,y,z$. One can do this so that the hexagon $Q(x,y,z)$ itself varies continuously in $\mathbb{H}^2$ as a function of $x,y,z$: fix an oriented line and a point on that line; go distance $x/2$; turn left and go distance $x'$; turn left and go distance $y/2$; etc.

Once you have the hexagon $Q(x,y,z)$, you can reflect across the $x'$ side to get another hexagon $Q'$, still depending continuously on $x,y,z$. Then, from the polygon $Q \cup Q'$, you can write down the matrix for translating its $y'$ side to its reflected $y'$ side and the matrix for translating its $z'$ side to its reflected $z'$ side; these matrices will depend continuously on $x,y,z$. And these matrices generate the discrete, convex, cocompact group such that the quotient of the convex hull of its limit set is the required pair of pants.