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.
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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)
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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
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DOI: https://doi.org/10.1007/s10455-012-9317-1