Calibrated Geometry in Hyperkähler Cones, 3-Sasakian Manifolds, and Twistor Spaces

We systematically study calibrated geometry in hyperkähler cones C⁴ⁿ⁺⁴ , their 3-Sasakian links M⁴ⁿ⁺³ , and the corresponding twistor spaces Z⁴ⁿ⁺² , emphasizing the relationships between submanifold geometries in various spaces. Our analysis highlights the role played by a canonical Sp(n)U(1) -structure γ on the twistor space Z . We observe that Re(e- i θ γ ) is an S¹ -family of semi-calibrations and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri–Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler–Einstein and nearly Kähler structures.