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Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

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Abstract

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S n by looking at the conormal bundle of appropriate submanifolds of S n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in R n in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G2 and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.

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Authors and Affiliations

  1. Mathematics Department, McMaster University, 1280, Main Street West, Hamilton, Ontario, L8S 4K1, Canada

    Spiro Karigiannis & Maung Min-Oo

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  1. Spiro Karigiannis
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  2. Maung Min-Oo
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Correspondence to Spiro Karigiannis.

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Mathematics Subject Classification (2000): 53-XX, 58-XX.

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Karigiannis, S., Min-Oo, M. Calibrated Subbundles in Noncompact Manifolds of Special Holonomy. Ann Glob Anal Geom 28, 371–394 (2005). https://doi.org/10.1007/s10455-005-1940-7

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  • DOI: https://doi.org/10.1007/s10455-005-1940-7

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