Generalized $\alpha$-attractor models from elementary hyperbolic surfaces

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.