We investigate the condensation of (K) over bar (0) mesons along with K- co
ndensation in neutrino-trapped matter with and without hyperons. Calculatio
ns are performed in the relativistic mean field models in which both the ba
ryon-baryon and (anti)kaon-baryon interactions are mediated by meson exchan
ge. In the neutrino-trapped matter relevant to protoneutron stars, the crit
ical density of K- condensation is shifted considerably to higher density w
hereas that of (K) over bar (0) condensation is shifted slightly to higher
density with respect to that of the neutrino-free case. The onset of K- con
densation always occurs earlier than that of (K) over bar (0) condensation.
A significant region of maximum mass protoneutron stars is found to contai
n (K) over bar (0) condensate for larger values of the antikaon potential.
With the appearance of (K) over bar (0) condensation. there is a region of
symmetric nuclear matter in the inner core of a protoneutron star. We note
that the gross properties of the protoneutron stars are significantly affec
ted by the antikaon condensation for larger values of the antikaon potentia
l. It is found that the maximum mass of a protoneutron star containing K- a
nd (K) over bar (0) condensate is greater than that of the corresponding ne
utron star. We reexamine the implication of this scenario in the context of
the metastability of protoneutron stars and their evolution to low mass bl
ack holes.