  Observations of Keplerian disks of masers in NCG 4258 and other Seyfert
galaxies can be used to obtain geometric distance estimates and derive the
Hubble constant. The ultimate precision of such measurements could be limited
by uncertainties in the disk geometry. Using a time-dependent linear theory
model, we study the evolution of a thin initially eccentric disk under
conditions appropriate to sub-pc scales in Active Galactic Nuclei. The
evolution is controlled by a combination of differential precession driven by
the disk potential and propagating eccentricity waves that are damped by
viscosity. A simple estimate yields a circularization timescale of
approximately 10 Myr at 0.1 pc. Numerical solutions for the eccentricity
evolution confirm that damping commences on this timescale, but show that the
subsequent decay rate of the eccentricity depends upon the uncertain strength
of viscous damping of eccentricity. If eccentricity waves are important further
decay of the eccentricity can be slow, with full circularization requiring up
to 50 Myr for disks at radii of 0.1 pc to 0.2 pc. Observationally, this implies
that it is plausible that enough time has elapsed for the eccentricity of
masing disks to have been substantially damped, but that it may not be
justified to assume vanishing eccentricity. We predict that during the damping
phase the pericenter of the eccentric orbits describes a moderately tightly
wound spiral with radius.
