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Topology of ϕ-convex domains in calibrated manifolds

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Abstract

In [5], Harvey and Lawson showed that for any calibration ϕ there is an integer bound for the homotopy dimension of a strictly ϕ-convex domain and constructed a method to get these domains by using ϕ-free submanifolds. Here, we show how to get examples of ϕ-free submanifolds with different homotopy types for the quaternion calibration in ℍn, associative calibration, and coassociative calibration in G 2 manifolds. Hence we give examples of strictly ϕ-convex domains with different homotopy types allowed by Morse Theory.

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References

  1. S. Akbulut and S. Salur. Calibrated Manifolds and Gauge Theory, math.GT/0402368.

  2. A. Andreotti and T. Frankel. The Lefschetz Theorem on Hyperplane Sections. Annals of Mathematics (2), 69 (1959), 713–717.

    Article  MathSciNet  MATH  Google Scholar 

  3. F.R. Harvey and R.O. Wells Jr. Zero Sets of Non-Negative Strictly Plurisubharmonic Functions. Mat. Ann., 201 (1973), 165–170.

    Article  MathSciNet  MATH  Google Scholar 

  4. F.R. Harvey and H.B. Lawson Jr. Calibrated Geometries. Acta Mathematica, 118 (1982), 47–157.

    Article  MathSciNet  Google Scholar 

  5. F.R. Harvey and H.B. Lawson Jr. Plurisubharmonic Functions In Calibrated Geometries, math.CV/0601484.

  6. F.R. Harvey and H.B. Lawson Jr. On Boundaries of Complex Analytic Varieties, I. Ann. of Math., 102 (1975), 223–290.

    Article  MathSciNet  MATH  Google Scholar 

  7. F.R. Harvey and H.B. Lawson Jr. Geometric Residue Theorems. Amer. J. Math., 117(4) (1995), 47–157.

    Article  MathSciNet  Google Scholar 

  8. D.D. Joyce. Compact Manifolds with Special Holonomy. Oxford University Press (2000).

  9. H.F. Lai. Characteristic classes of real manifolds immersed in complex manifolds. Trans. Amer. Math. Soc., 172 (1972), 1–33.

    Article  MathSciNet  Google Scholar 

  10. J. Milnor. Morse Theory. Annals of Mathematics Studies, 51. Princeton, NJ: Princeton Univ. Press (1963).

    Google Scholar 

  11. K. Oka. Domaines pseudoconvexes. Tohoku Math. J., 49 (1942), 15–52.

    MathSciNet  MATH  Google Scholar 

  12. S. Salur and O. Santillan. Mirror Duality via G 2 and Spin (7) Manifolds, math.DG/07071356.

  13. A. Strominger, S-T. Yau and E. Zaslow. Mirror Symmetry is T-Duality. Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 333–347, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI (2001).

    Google Scholar 

  14. G. Tian. Gauge Theory and Calibrated Geometry I. Ann. of Math. (2) 151(1) (2000), 193–268.

    Article  MathSciNet  MATH  Google Scholar 

  15. S.M. Webster. Minimal Surfaces in a Kähler Surface. J. Differential Geom., 20(2) (1984), 463–470.

    MathSciNet  MATH  Google Scholar 

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Author notes

  1. Ibrahim Unal

    Present address: Northern Cyprus Campus, Middle East Technical University, Kalkanli, Guzelyurt, KKTC, Mersin 10, Turkey

Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Ibrahim Unal

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  1. Ibrahim Unal
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Correspondence to Ibrahim Unal.

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Unal, I. Topology of ϕ-convex domains in calibrated manifolds. Bull Braz Math Soc, New Series 42, 259–275 (2011). https://doi.org/10.1007/s00574-011-0014-7

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  • DOI: https://doi.org/10.1007/s00574-011-0014-7

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