Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$

We construct a deformation space for minimal immersions into $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$ with its holomorphic Killing form, and define an explicit, mapping class group invariant, complex hyperKähler structure on such deformation space. Such complex hyperKähler structure restricts to: Donaldson’s hyperKähler structure on a totally real copy of almost-Fuchsian space, Mazzoli-Seppi-Tamburelli’s para-hyperKähler structure on a totally real copy of maximally globally hyperbolic $\mathbb{A}\mathrm{d}{\mathbb{S}}^3$ manifolds, and Loustau-Sanders’ complex bi-Kähler structure on a holomorphic copy of quasi-Fuchsian space. Moreover, all these loci are totally geodesic. We discuss other geometric aspects of this space, such as the Hamiltonian property of a natural ${\mathbb{C}}^{\ast }$-action, involving the Goldman symplectic form and a complex-valued area functional.