  The left-right symmetric model (LRSM), originally proposed to explain parity
violation in low energy processes, has since emerged as an attractive framework
for light neutrino masses via the seesaw mechanism. The scalar sector of the
minimal LRSM consists of an $SU(2)$ bi-doublet, as well as left- and
right-handed weak isospin triplets, thus making the corresponding vacuum
structure much more complicated than that of the Standard Model. In particular,
the desired ground state of the Higgs potential should be a charge conserving,
and preferably global, minimum with parity violation at low scales. We show
that this is not a generic feature of the LRSM potential and happens only for a
small fraction of the parameter space of the potential. We also analytically
study the potential for some simplified cases and obtain useful conditions
(though not necessary) to achieve successful symmetry breaking. We then carry
out a detailed statistical analysis of the minima of the Higgs potential using
numerical minimization and find that for a large fraction of the parameter
space, the potential does not have a good vacuum. Imposing the analytically
obtained conditions, we can readily find a small part of the parameter space
with good vacua. Consequences for some scalar masses are also discussed.
