  Collapse in dipolar Bose-Einstein condensates may be arrested by quantum
fluctuations. Due to the anisotropy of the dipole-dipole interactions, the
dipole-driven collapse induced by soft excitations is compensated by the
repulsive Lee-Huang-Yang contribution resulting from quantum fluctuations of
hard excitations, in a similar mechanism as that recently proposed for
Bose-Bose mixtures. The arrested collapse results in self-bound filament-like
droplets, providing an explanation to recent dysprosium experiments. Arrested
instability and droplet formation are novel general features directly linked to
the nature of the dipole-dipole interactions, and should hence play an
important role in all future experiments with strongly dipolar gases.
