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

Abstract

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 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 hyperkähler 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 Kähler-Einstein and nearly-Kähler structures.

55 pages, 1 figure. Version 3: Minor revisions and clarifications following referee's report. Final version, to appear in Canadian Journal of Mathematics

References in corpus (2)

• Special Lagrangians in nearly Kähler ${\mathbb{CP}}^3$
• Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds