  We analyze the stability and dynamics of toroidal liquid droplets. In
addition to the Rayleigh instabilities akin to those of a cylindrical droplet
there is a shrinking instability that is unique to the topology of the torus
and dominates in the limit that the aspect ratio is near one (fat tori). We
first find an analytic expression for the pressure distribution inside the
droplet. We then determine the velocity field in the bulk fluid, in the Stokes
flow regime, by solving the biharmonic equation for the stream function. The
flow pattern in the external fluid is analyzed qualitatively by exploiting
symmetries. This elucidates the detailed nature of the shrinking mode and the
swelling of the cross-section following from incompressibility. Finally the
shrinking rate of fat toroidal droplets is derived by energy conservation.
