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Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy

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Abstract

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mn i , of riemannian manifolds with special holonomy. We show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension ≥4. Additional results on the the detailed structure of the singular set support our main conjecture that if the Mn i are compact and a certain characteristic number, C(Mn i ), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Jeff Cheeger

  2. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Gang Tian

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Gang Tian

Authors
  1. Jeff Cheeger
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  2. Gang Tian
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Correspondence to Jeff Cheeger.

Additional information

Communicated by P. Sarnak

The first author was partially supported by NSF Grant DMS 0104128 and the second by NSF Grant DMS 0302744.

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Cheeger, J., Tian, G. Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy. Commun. Math. Phys. 255, 391–417 (2005). https://doi.org/10.1007/s00220-004-1279-0

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