  We investigate the liquid-gas phase transition of hot protoneutron stars
shortly after their birth following supernova explosion and the composition and
structure of hyperon-rich (proto)neutron stars within a relativistic mean-field
model where the nuclear symmetry energy has been constrained from the measured
neutron skin thickness of finite nuclei. Light clusters are abundantly formed
with increasing temperature well inside the neutrino-sphere for an uniform
supernova matter. Liquid-gas phase transition is found to suppress the cluster
yield within the coexistence phase as well as decrease considerably the
neutron-proton asymmetry over a wide density range. We find symmetry energy has
a modest effect on the boundaries and the critical temperature for the
liquid-gas phase transition, and the composition depends more sensitively on
the number of trapped neutrinos and temperature of the protoneutron star. The
influence of hyperons in the dense interior of stars makes the overall equation
of state soft. However, neutrino trapping distinctly delays the appearance of
hyperons due to abundance of electrons. We also find that a softer symmetry
energy further makes the onset of hyperon less favorable. The resulting
structures of the (proto)neutron stars with hyperons and with liquid-gas phase
transition are discussed.
