Shape polarization and coexistence of high-$K$ three-quasiparticle states in odd-mass $N=106$ isotones

Three-quasiparticle $K$-isomeric states in odd-mass $N=106$ isotones within the $A\sim 180$ mass region are systematically investigated using configuration-constrained potential energy surface calculations. The calculations successfully reproduce the excitation energies and deformations of known high-$K$ isomers in the nuclei from $^{175}$Tm to $^{181}$Re. For the nuclei closer to the $Z=82$ shell closure ($^{183}$Ir, $^{185}$Au, and $^{187}$Tl), predictions for the configurations of observed and yet-to-be-observed isomers are provided. The results reveal strong shape polarization, where the three-quasiparticle states are driven to larger deformations compared to the often shape-soft or spherical ground states. A particularly rich spectrum of shape coexistence is predicted in $^{187}$Tl, where several high-$K$ three-quasiparticle configurations with distinct prolate, oblate, and triaxial shapes are found to coexist at similar excitation energies. Notably, the oblate-deformed $K^π=29/2^+$ configuration at $E_x = 1839$ keV is proposed to be responsible for a long-lived isomer. This study provides a comprehensive picture of shape evolution and coexistence in high-$K$ multi-quasiparticle states, offering valuable insights for future experimental research.