Cantor families of periodic solutions for spherical capillary water waves

The spherical capillary water waves system is considered on the sphere, with small-amplitude time-periodic solutions as the objects of interest. Periodic-solution conjecture. There is a Cantor family of small amplitude periodic solutions to the spherical capillary water waves system. This conjecture is motivated by hydrodynamical experiments under zero gravity and by the existence of standing gravity-capillary water waves. Its validity for spherical capillary water waves remains open.