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?**