  We have investigated the ground-state structures of even-even
neutron-deficient isotopes of Hg and Pb nuclei within the framework of a
deformed relativistic mean field formalism for a number of commonly used
parameter sets, namely NL1, NL3, NL-SH, NL-RA1 and TM1. The ground states of a
bunch of Hg and Pb-isotopes towards the proton-dripline are found to be
deformed for all the forces, with a constant pairing gap. The small differences
in the ground- and the first-excited state binding energies predict a sea of
low-lying excited states, and hence of shape co-existence, for both the Hg and
Pb nuclei. In general, a discrepancy between the experimentally observed and
theoretically predicted (sign of) quadrupole deformations is noticed in this
mass region. The constrained potential energy surfaces and the single-particle
energy spectra analyzed for some selected nuclei show that the known large
shell gap for Pb nuclei at Z=82 is almost extinguished for its proton-rich
isotopes.
