Hyperplane absolute winning property of bounded orbits under diagonalizable flows on ${\mathrm{S}\mathrm{L}}_3(\mathbb{C})\mathrm{/}{\mathrm{S}\mathrm{L}}_3({\mathcal{O}}_{\mathbb{K}})$

We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on ${\mathrm{S}\mathrm{L}}_3(\mathbb{R})\mathrm{/}{\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ to ${\mathrm{S}\mathrm{L}}_3(\mathbb{C})\mathrm{/}{\mathrm{S}\mathrm{L}}_3({\mathcal{O}}_{\mathbb{K}})$, where $\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski’s Linear Forms Theorem. We then set up an appropriate Schmidt game in ${\mathbb{C}}^3$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in ${\mathbb{C}}^3$ relative to an approximation by imaginary quadratic rationals in $\mathbb{K}$.