We implement variational techniques and an implicit function theorem to derive constraints on angular velocity under which we may verify the existence, boundary regularity, and stability of an energyminimising family of rotating liquid drops in a neighbourhood of the closed unit ball in R n+1 .

We establish sufficient conditions for uniqueness in the context of an energy minimisation property derived earlier by the author for rotating liquid drops of arbitrary dimension. In particular, we obtain unique, global solutions of an associated geometric evolution equation whenever appropriate restrictions are placed on an initial condition corresponding to a fixed angular velocity. These solutions are demonstrated to converge smoothly to a known stable minimal equilibrium, and we prove that the boundary of each such energy minimiser is uniquely determined in a Lipschitz neighbourhood of the unit sphere.

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