  We consider generalized $\alpha$-attractor models whose scalar potentials are
globally well-behaved and whose scalar manifolds are elementary hyperbolic
surfaces. Beyond the Poincar\'e disk $\mathbb{D}$, such surfaces include the
hyperbolic punctured disk $\mathbb{D}^\ast$ and the hyperbolic annuli
$\mathbb{A}(R)$ of modulus $\mu=2\log R>0$. For each elementary surface, we
discuss its decomposition into canonical end regions and give an explicit
construction of the embedding into the Kerekjarto-Stoilow compactification
(which in all cases is the unit sphere), showing how this embedding allows for
a universal treatment of globally well-behaved scalar potentials upon expanding
their extension in real spherical harmonics. For certain simple but natural
choices of extended potentials, we compute scalar field trajectories by
projecting numerical solutions of the lifted equations of motion from the
Poincar\'e half-plane through the uniformization map, thus illustrating the
rich cosmological dynamics of such models.
