  Most extrasolar planets are observed to have eccentricities much larger than
those in the solar system. Some of these planets have sibling planets, with
comparable masses, orbiting around the same host stars. In these multiple
planetary systems, eccentricity is modulated by the planets' mutual secular
interaction as a consequence of angular momentum exchange between them. For
mature planets, the eigenfrequencies of this modulation are determined by their
mass and semi-major axis ratios. But, prior to the disk depletion, self gravity
of the planets' nascent disks dominates the precession eigenfrequencies. We
examine here the initial evolution of young planets' eccentricity due to the
apsidal libration or circulation induced by both the secular interaction
between them and the self gravity of their nascent disks. We show that as the
latter effect declines adiabatically with disk depletion, the modulation
amplitude of the planets' relative phase of periapse is approximately invariant
despite the time-asymmetrical exchange of angular momentum between planets.
However, as the young planets' orbits pass through a state of secular
resonance, their mean eccentricities undergo systematic quantitative changes.
For applications, we analyze the eccentricity evolution of planets around
Upsilon Andromedae and HD168443 during the epoch of protostellar disk
depletion. We find that the disk depletion can change the planets' eccentricity
ratio. However, the relatively large amplitude of the planets' eccentricity
cannot be excited if all the planets had small initial eccentricities.
