The results of an investigation on the stability of a doubly connected axisymmetric equilibrium free surface pinned to the edges of two coaxial disks are presented. The general boundary of the region where the interface is stable is constructed in the plane of the parameters determining the slenderness of a liquid bridge and its relative volume. Surface forces and arbitrary (not only axisymmetric) perturbations are taken into account. The general boundary of the stability region was calculated completely in the past only for a weightless fixed-contactline liquid bridge between equal disks. The influence of axially directed gravity, isorotation and disks inequality on the evolution of this boundary has been analyzed successively. As a result, the families of the stability boundaries have been obtained for fixed-contact-line liquid bridges between equal disks in a wide range of Bond numbers, for isorotating weightless bridges between equal disks, in a wide range of Weber numbers and for weightless fixed-contact-line liquid bridges when the disk radii ratio is varied. Basing on the solution of the bifurcation problem for the critical equilibrium states, the conclusion on the results of stability losing has been made for starting system of fixed-contact line weightless bridge between equal-disks. The stability of the melt during crystal growth using the floating zone technique can be considered as a special case of the presented results. Finally as an example, for a crystal growth system using the Stepanov's method the effect of free surface unconnectivity on the stability has been investigated under zero gravity conditions
