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.
Categories: math.DG