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

TL;DR

This paper explores calibrated geometries in hyperk"ahler cones, 3-Sasakian manifolds, and twistor spaces, revealing new characterizations of special submanifolds and their relationships across these geometries.

Contribution

It introduces new characterizations of complex Lagrangian and isotropic cones in hyperk"ahler cones and extends existing theorems about submanifolds in twistor spaces.

Findings

New characterizations of complex Lagrangian cones

Generalization of Storm's theorem on twistor submanifolds

Analysis of semi-calibrations via $ ext{Re}(e^{- i heta} \, \gamma)$

Abstract

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.