Bulk and decay properties of neutron-deficient odd-mass Hg isotopes near <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>185</mml:mn></mml:mrow></mml:math>

Ground and isomeric states of the neutron-deficient odd-$A$ isotopes $^{183}$Hg, $^{185}$Hg, $^{187}$Hg are described from a microscopic calculation based on self-consistent, axially-deformed Hartree-Fock mean field with Skyrme functional pairing within BCS approximation. For each equilibrium shape different odd-neutron states, results square charge radii magnetic dipole moments given analyzed in context their sensitivity to nuclear deformation spin parity. Spin-isospin correlations proton-neutron quasiparticle random phase approximation then introduced obtain distributions Gamow-Teller strength $\beta^+/EC$ half-lives these isotopes, whose measurements planned at ISOLDE-CERN using total absorption gamma-ray spectroscopy techniques.