Deformations of calibrated subbundles in noncompact manifolds of special holonomy via twisting by special sections

Abstract

We study special Lagrangian submanifolds in the Calabi-Yau manifold $T^*S^n$ with the Stenzel metric, as well as calibrated submanifolds in the $\text{G}_2$-manifold $\Lambda^2_-(T^*X)$ $(X^4 = S^4, \mathbb{CP}^2)$ and the $\text{Spin}(7)$-manifold $\$_{\!-}(S^4)$, both equipped with the Bryant-Salamon metrics. We twist naturally defined calibrated subbundles by sections of the complementary bundles and derive conditions for the deformations to be calibrated. We find that twisting the conormal bundle $N^*L$ of $L^q \subset S^n$ by a $1$-form $\mu \in \Omega^1(L)$ does not provide any new examples because the Lagrangian condition requires $\mu$ to vanish. Furthermore, we prove that the twisted bundles in the $\text{G}_2$- and $\text{Spin}(7)$-manifolds are associative (coassociative) and Cayley, respectively, if the base is minimal (negative superminimal) and the section holomorphic (parallel). This demonstrates that the (co-)associative and Cayley subbundles allow deformations destroying the linear structure of the fiber, while the base space remains of the same type after twisting. While the results for the two spaces of exceptional holonomy are in line with the findings in Euclidean spaces established by Karigiannis and Leung (2012), the special Lagrangian bundle construction in $T^*S^n$ is much more rigid than in the case of $T^*\mathbb{R}^n$.