Abstract
We extend the ‘bundle constructions’ of calibrated submanifolds, due to Harvey–Lawson in the special Lagrangian case, and to Ionel–Karigiannis–Min-Oo in the cases of exceptional calibrations, by ‘twisting’ the bundles by a special (harmonic, holomorphic, or parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these ‘calibrated subbundles’ includes deformations which destroy the linear structure of the fibre.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borisenko, A.: Ruled special Lagrangian surfaces. In: Minimal Surfaces, pp. 269–285, Advances in Soviet Mathematics, vol. 15. American Mathematical Society, Providence (1993)
Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Ionel M., Karigiannis S., Min-Oo M.: Bundle constructions of calibrated submanifolds in \({\mathbb {R}^7}\) and \({\mathbb {R}^8}\) . Math. Res. Lett. 12, 493–512 (2005)
Joyce D.: Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces. Ann. Global Anal. Geom. 25, 301–352 (2004)
Joyce D., Salur S.: Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary. Geom. Topol. 9, 1115–1146 (2005)
Karigiannis S., Min-Oo M.: Calibrated subbundles in noncompact manifolds of special holonomy. Ann. Global Anal. Geom. 28, 371–394 (2005)
Lockhart R.: Fredholm, Hodge and Liouville theorems on noncompact manifolds. Trans. Amer. Math. Soc. 301, 1–35 (1987)
Lockhart R.B., McOwen R.C.: Elliptic differential operators on noncompact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 12, 409–447 (1985)
Lotay J.: Deformation theory of asymptotically conical coassociative 4-folds. Proc. London Math. Soc. (3) 99, 386–424 (2009)
Lotay J.: Asymptotically conical associative 3-folds. Q. J. Math. 62, 131–156 (2011)
McLean R.C.: Deformations of calibrated submanifolds. Comm. Anal. Geom. 6, 705–747 (1998)
Pacini T.: Deformations of asymptotically conical special Lagrangian submanifolds. Pacific J. Math. 215, 151–181 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karigiannis, S., Leung, N.CH. Deformations of calibrated subbundles of Euclidean spaces via twisting by special sections. Ann Glob Anal Geom 42, 371–389 (2012). https://doi.org/10.1007/s10455-012-9317-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-012-9317-1