Hexagonal boron nitride supports phonon polaritons in its two Reststrahlen bands. In this paper, we investigate phonon polaritons in cylindrically curved hexagonal boron nitride thin films. The phonon polariton modes in such structure carry orbital angular momentums depending on its azimuthal index. For extremely small-size structures, high order polariton modes show cutoff behaviors; while, for large-size ones, modes with low azimuthal indexes are nearly degenerate, showing similar mode effective indexes. In dimer structures, phonon polariton modes in the neighboring structures are coupled, creating hybrid modes; gap phonon polaritons arise due to such coupling. For large-size dimers, multiple gap phonon polariton modes have been found. Then, cylindrically curved hexagonal boron nitride thin film is placed on a substrate, which also leads to the emergence of multiple gap phonon polariton modes near the touching point. In the end, we vary the geometric parameters of the structures and give some discussions about the phonon polariton modes. Based on these investigations, we may say that the curvature can strongly affect the phonon polariton modes in h-BN thin films.