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

We systematically study calibrated geometry in hyperk\"ahler cones $C^{4n+4}$, their 3-Sasakian links $M^{4n+3}$, and the corresponding twistor spaces $Z^{4n+2}$, emphasizing the relationships between submanifold geometries in various spaces. Our analysis emphasizes the role played by a canonical $\mathrm{Sp}(n)\mathrm{U}(1)$-structure $\gamma$ on the twistor space $Z$. We observe that $\mathrm{Re}(e^{- i \theta} \gamma)$ is an $S^1$-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 hyperk\"{a}hler cones, generalizing a result of Ejiri and Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the K\"{a}hler-Einstein and nearly-K\"{a}hler structures.