  In this note, we propose several unsolved problems concerning the
irrotational oscillation of a water droplet under zero gravity. We will derive
the governing equation of this physical model, and convert it to a quasilinear
dispersive partial differential equation defined on the sphere, which formally
resembles the capillary water waves equation but describes oscillation defined
on curved manifold instead. Three types of unsolved mathematical problems
related to this model will be discussed in observation of hydrodynamical
experiments under zero gravity: (1) Strichartz type inequalities for the
linearized problem (2) existence of periodic solutons (3) normal form reduction
and generic lifespan estimate. It is pointed out that all of these problems are
closely related to certain Diophantine equations, especially the third one.
