Deformations of calibrated subbundles in noncompact manifolds of special holonomy via twisting by special sections
Romy Marie Merkel
TL;DR
This work extends the study of calibrated subbundles to complete, noncompact manifolds with special holonomy by twisting subbundles with sections of complementary bundles. It establishes that in the Calabi–Yau Stenzel setting, twisting N^*L by a 1-form μ cannot yield new special Lagrangian submanifolds since μ must vanish. In Bryant–Salamon G2 and Spin(7) geometries, twisting by holomorphic or parallel sections preserves calibration precisely when the base L^2 is minimal (associative and Cayley cases) or negative superminimal (coassociative case); these twists deform the fiber while preserving the base’s calibrated type. Explicit G2 examples show that associative submanifolds can arise from holomorphic twists of Veronese-type bases, while the equatorial sphere yields no new examples due to the absence of nontrivial holomorphic sections. Overall, the results align with Euclidean analogs but reveal rigidity for special Lagrangian twists in the nonflat setting, and they highlight the nuanced interplay between base geometry and fiber twisting in calibrated submanifolds.
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 $Λ^2_-(T^*X)$ $(X^4 = S^4, \mathbb{CP}^2)$ 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 $μ\in Ω^1(L)$ does not provide any new examples because the Lagrangian condition requires $μ$ 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 $T^*\mathbb{R}^n$.