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

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?

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.

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.

• As far as I can tell you're showing that the Teichmuller space is in fact $(\mathbb{R}^+)^3$. But how does this give you a smooth Riemannian metric on $P$ varying continuously (in the smooth topology) with respect to $x,y,z$? – Pablo Lessa Jul 16 '14 at 15:29
• Ah, you're right, I have still missed the main point, which is to pick a continuously varying family of diffeomorphisms from a base hexagon $Q(x_0,y_0,z_0)$ to each of the other hexagons in this family……. – Lee Mosher Jul 16 '14 at 15:38