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Siberian Mathematical Journal

, Volume 56, Issue 6, pp 1093–1100 | Cite as

A geometric flow in the space of G 2-structures on the cone over S 3×S 3

  • Kh. Zh. KozhasovEmail author
Article
  • 49 Downloads

Abstract

We consider a flow of G 2-structures on a 7-dimensional manifold admitting a G 2-structure. The general solution to this flow is found in the case when the manifold is the cone over S 3×S 3. Weprove the convergence of the metric associated with the solution to the conical metric modulo homotheties.

Keywords

G2-structure G2-manifold flow of G2-structures cone over S3×S3 

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References

  1. 1.
    Yau S.-T., “On Calabi’s conjecture and some new results in algebraic geometry,” Proc. Nat. Acad. Sci. USA, 74, No. 5, 1798–1799 (1977).zbMATHCrossRefGoogle Scholar
  2. 2.
    Bryant R. L. and Salamon S. L., “On the construction of some complete metrics with exceptional holonomy,” Duke Math. J., 58, No. 3, 829–850 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Gibbons G. W., Page D. N., and Pope C. N., “Einstein metrics on S 3, R 3, and R 4 bundles,” Comm. Math. Phys., 127, No. 3, 529–553 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Joyce D. D., “Compact Riemannian 7-manifolds with holonomy G 2. I,” J. Differential Geom., 43, No. 2, 291–328 (1996).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Kovalev A. G., “Twisted connected sums and special Riemannian holonomy,” J. Reine Angew. Math., 565, 125–160 (2003).zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bryant R. L., “Some remarks on G 2-structures,” Proc. Gokova Geometry–Topology Conf., 2005, pp. 75–109.Google Scholar
  7. 7.
    Karigiannis S., “Flows of G 2-structures. I,” Quart. J. Math., 60, 487–522 (2009).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bryant R. L. and Xu Feng, “Laplacian flow for closed G 2-structures: Short Time Behavior,” arXiv article, 11 Jan 2011, arXiv:math.DG/1101.2004v1.Google Scholar
  9. 9.
    Grigorian S., “Short-time behavior of a modified Laplacian coflow of G 2-structures,” Adv. Math., 248, 378–415 (2013).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Joyce D. D., Compact Manifolds with Special Holonomy, Oxford Univ. Press, Oxford (2000) (Oxford Math. Monographs).Google Scholar
  11. 11.
    Bonan E., “Sur les variétés Riemanniennes à groupe d’holonomie G 2 ou Spin(7),” C. R. Acad. Sci. Paris, 262, 127–129 (1966).zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fernández M. and Gray A., “Riemannian manifolds with structure group G 2,” Ann. Mat. Pura Appl., 132, No. 4, 19–45 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Brandhuber A., Gomis J., Gubser S. S., and Gukov S., “Gauge theory at large N and new G 2 holonomy metrics,” Nucl. Phys. B, 611, No. 1–3, 179–204 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bazaĭkin Ya. V. and Bogoyavlenskaya O. A., “Complete Riemannian metrics with holonomy group G 2 on deformations of cones over S 3 × S 3,” Math. Notes, 93, No. 5, 643–653 (2013).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly

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