Nonequatorial scalar rings supported by rapidly spinning Gauss-Bonnet black holes

Black-hole spacetimes that possess stationary equatorial matter rings are known to exist in general relativity. We here reveal the existence of black-hole spacetimes that support nonequatorial matter rings. In particular, it is proved that rapidly rotating Kerr black holes in the dimensionless large-spin regime $\overline{a}>{\overline{a}}_{\mathrm{crit}}=\sqrt{{7+\sqrt{7}\mathrm{cos}[\frac{1}{3}\mathrm{arctan}(3\sqrt{3})]\ensuremath{-}\sqrt{21}\mathrm{sin}[\frac{1}{3}\mathrm{arctan}(3\sqrt{3})]}/12}\ensuremath{\simeq}0.78$ can support a pair of nonequatorial massive scalar rings that are negatively coupled to the Gauss-Bonnet curvature invariant of the spinning spacetime (here $\overline{a}\ensuremath{\equiv}J/{M}^{2}$ is the dimensionless angular momentum of the central supporting black hole). We explicitly prove that these nonequatorial scalar rings are characterized by the dimensionless functional relation $\ensuremath{-}\frac{57+28\sqrt{21}\mathrm{cos}[\frac{1}{3}\mathrm{arctan}(\frac{1}{3\sqrt{3}})]}{8(1+\sqrt{1\ensuremath{-}{\overline{a}}^{2}}{)}^{6}}\ifmmode\cdot\else\textperiodcentered\fi{}\frac{\overline{\ensuremath{\eta}}}{{\overline{\ensuremath{\mu}}}^{2}}\ensuremath{\rightarrow}{1}^{+}$ in the large-mass $\overline{\ensuremath{\mu}}\ensuremath{\equiv}M\ensuremath{\mu}\ensuremath{\gg}1$ regime (here ${\overline{\ensuremath{\eta}}<0,\ensuremath{\mu}}$ are, respectively, the nontrivial coupling parameter of the composed Einstein-Gauss-Bonnet-massive-scalar field theory and the proper mass of the supported nonminimally coupled scalar field).