$\mathcal{ F}$ of isometry classes of complete metrics of constant curvature -1, that is, hyperbolic metrics on the surface [S]{}. Define an involution $r:
{\sf S}\rightarrow {\sf S}$ by $r(e^{i\theta},t)
=(e^{-i\theta},t)$. Let $\mathcal{ F}_0 \subset \mathcal{  F}$ be the set of isometry classes in $\mathcal{  F}$ for which there exists a representative surface on which $r$ is an isometry. Henceforth we shall treat elements of $\mathcal{  F}$ and $\mathcal{  F}_0$ as hyperbolic surfaces and suppose, in the latter case, that $r$ is an isometry. A surface in $\mathcal{  F}$ is called a [*flute*]{}; one in $\mathcal{  F}_0$ is called an [*untwisted flute*]{}. An explicit construction of flutes is given in [@bas1]. A flute has one infinite end and has a finite end associated to each of the deleted points $(1,n)$ and to the ideal boundary ${\rm S}^1\times\{0\}.$ Note that each of these ends can be of the first or the second kind, depending on the hyperbolic metric we have chosen. The proposition below, whose proof is given after we develop some notation, shows that this definition of a flute surface is consistent with the definition given in the introduction.

\[planar\] $S$ is a flute surface if and only if $S$ 