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Calibrated geometry in hyperkähler cones, 3-Sasakian manifolds, and twistor spaces
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 01 April 2024, pp. 1-59
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We systematically study calibrated geometry in hyperkähler 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 highlights 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 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.
