Duke Mathematical Journal

Volume-minimizing cycles in Grassmann manifolds

Herman Gluck, Dana Mackenzie, and Frank Morgan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J., Volume 79, Number 2 (1995), 335-404.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077285156

Digital Object Identifier
doi:10.1215/S0012-7094-95-07909-5

Mathematical Reviews number (MathSciNet)
MR1344765

Zentralblatt MATH identifier
0837.53035

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 57R20: Characteristic classes and numbers 58E99: None of the above, but in this section

Citation

Gluck, Herman; Mackenzie, Dana; Morgan, Frank. Volume-minimizing cycles in Grassmann manifolds. Duke Math. J. 79 (1995), no. 2, 335--404. doi:10.1215/S0012-7094-95-07909-5. https://projecteuclid.org/euclid.dmj/1077285156


Export citation

References

  • [A] F. J. Almgren, Jr., $Q$-valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc. 8 (1983), no. 2, 327–328.
    Mathematical Reviews (MathSciNet): MR84b:49052
    Zentralblatt MATH: 0557.49021
    Digital Object Identifier: doi:10.1090/S0273-0979-1983-15106-6
    Project Euclid: euclid.bams/1183550129
  • [B] M. Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4) 5 (1972), 1–44.
    Mathematical Reviews (MathSciNet): MR46:8119
    Zentralblatt MATH: 0227.52005
  • [BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958), 458–538.
    Mathematical Reviews (MathSciNet): MR21:1586
    Zentralblatt MATH: 0097.36401
    Digital Object Identifier: doi:10.2307/2372795
    JSTOR: links.jstor.org
  • [B1] K. A. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1 (1991), no. 4, 329–338.
    Mathematical Reviews (MathSciNet): MR92k:49082
    Zentralblatt MATH: 0738.49022
  • [B2] K. A. Brakke, Soap films and covering spaces, preprint, 1993.
    Mathematical Reviews (MathSciNet): MR1393090
    Zentralblatt MATH: 0848.49025
    Digital Object Identifier: doi:10.1007/BF01895675
  • [CG] E. Calabi and H. Gluck, What are the best almost-complex structures on the $6$-sphere? Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 2, vol. 54, Amer. Math. Soc., Providence, 1993, pp. 99–106.
    Mathematical Reviews (MathSciNet): MR94k:53047
    Zentralblatt MATH: 0790.53036
  • [dR] G. de Rham, On the area of complex manifolds, Notes for the Seminar on Several Complex Variables, 1957-58, Institute for Advanced Study, Princeton.
  • [DGGW1] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, You cannot hear the mass of a homology class, Comment. Math. Helv. 64 (1989), no. 4, 589–617.
    Mathematical Reviews (MathSciNet): MR90k:58233
    Zentralblatt MATH: 0694.53037
    Digital Object Identifier: doi:10.1007/BF02564696
  • [DGGW2] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, Conformal isospectral deformations, Indiana Univ. Math. J. 41 (1992), no. 1, 99–107.
    Mathematical Reviews (MathSciNet): MR93c:58218
    Zentralblatt MATH: 0742.58055
    Digital Object Identifier: doi:10.1512/iumj.1992.41.41006
  • [DGGW3] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, The geometry of isospectral deformations, Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 3, vol. 54, Amer. Math. Soc., Providence, 1993, pp. 135–154.
    Mathematical Reviews (MathSciNet): MR94b:58096
    Zentralblatt MATH: 0811.53035
  • [DGGW4] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, The inaudible geometry of nilmanifolds, Invent. Math. 111 (1993), no. 2, 271–284.
    Mathematical Reviews (MathSciNet): MR93k:58222
    Zentralblatt MATH: 0779.53026
    Digital Object Identifier: doi:10.1007/BF01231288
  • [F1] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969.
    Mathematical Reviews (MathSciNet): MR41:1976
    Zentralblatt MATH: 0176.00801
  • [F2] H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351–407.
    Mathematical Reviews (MathSciNet): MR50:1095
    Zentralblatt MATH: 0289.49044
    Digital Object Identifier: doi:10.1512/iumj.1974.24.24031
  • [F3] H. Federer, Some theorems on integral currents, Trans. Amer. Math. Soc. 117 (1965), 43–67.
    Mathematical Reviews (MathSciNet): MR29:5984
    Zentralblatt MATH: 0136.18204
    Digital Object Identifier: doi:10.2307/1994196
  • [GMZ] H. Gluck, F. Morgan, and W. Ziller, Calibrated geometries in Grassmann manifolds, Comment. Math. Helv. 64 (1989), no. 2, 256–268.
    Mathematical Reviews (MathSciNet): MR90h:53077
    Zentralblatt MATH: 0681.53039
    Digital Object Identifier: doi:10.1007/BF02564674
  • [GW] H. Gluck and F. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983), no. 1, 107–132.
    Mathematical Reviews (MathSciNet): MR84g:53056
    Zentralblatt MATH: 0523.55020
    Digital Object Identifier: doi:10.1215/S0012-7094-83-05003-2
    Project Euclid: euclid.dmj/1077303001
  • [GWY] H. Gluck, F. Warner, and C. T. Yang, Division algebras, fibrations of spheres by great spheres, and the topological determination of space by the gross behavior of its geodesics, Duke Math. J. 50 (1983), no. 4, 1041–1076.
    Mathematical Reviews (MathSciNet): MR85i:53047
    Zentralblatt MATH: 0534.53039
    Digital Object Identifier: doi:10.1215/S0012-7094-83-05044-5
    Project Euclid: euclid.dmj/1077303489
  • [GWZ1] H. Gluck, F. Warner, and W. Ziller, The geometry of the Hopf fibrations, Enseign. Math. (2) 32 (1986), no. 3-4, 173–198.
    Mathematical Reviews (MathSciNet): MR88e:53067
    Zentralblatt MATH: 0616.53038
  • [GWZ2] H. Gluck, F. Warner, and W. Ziller, Fibrations of spheres by parallel great spheres and Berger's Rigidity Theorem, Ann. Global Anal. Geom. 5 (1987), no. 1, 53–82.
    Mathematical Reviews (MathSciNet): MR89e:53063
    Zentralblatt MATH: 0642.53046
    Digital Object Identifier: doi:10.1007/BF00140754
  • [GZ] H. Gluck and W. Ziller, On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986), no. 2, 177–192.
    Mathematical Reviews (MathSciNet): MR87j:53063
    Zentralblatt MATH: 0605.53022
    Digital Object Identifier: doi:10.1007/BF02621910
  • [GHV] W. Greub, S. Halperin, and R. Vanstone, Connections, Curvature, and Cohomology, Academic Press, New York, 1976.
    Mathematical Reviews (MathSciNet): MR53:4110
    Zentralblatt MATH: 0372.57001
  • [Gu] W. Gu, The stable $4$-dimensional geometry of the real Grassmann manifolds, Ph.D. thesis, Univ. of Pennsylvania, 1995.
  • [H1] R. Harvey, Calibrated geometries, Proc. Internat. Cong. Math., Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 797–808.
    Mathematical Reviews (MathSciNet): MR87d:53123
    Zentralblatt MATH: 0565.53032
  • [H2] R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Boston, 1990.
    Mathematical Reviews (MathSciNet): MR91e:53056
    Zentralblatt MATH: 0694.53002
  • [HL] R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157.
    Mathematical Reviews (MathSciNet): MR85i:53058
    Zentralblatt MATH: 0584.53021
    Digital Object Identifier: doi:10.1007/BF02392726
  • [L1] G. Lawlor, A sufficient criterion for a cone to be area-minimizing, Mem. Amer. Math. Soc. 91 (1991), no. 446, vi+111.
    Mathematical Reviews (MathSciNet): MR92d:49079
    Zentralblatt MATH: 0745.49029
  • [L2] G. Lawlor, Proving area minimization by slicing, preprint, 1993.
  • [LM] G. Lawlor and F. Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166 (1994), no. 1, 55–83.
    Mathematical Reviews (MathSciNet): MR95i:58051
    Zentralblatt MATH: 0830.49028
    Project Euclid: euclid.pjm/1102621245
  • [M1] F. Morgan, Area-minimizing surfaces, faces of Grassmannians, and calibrations, Amer. Math. Monthly 95 (1988), no. 9, 813–822.
    Mathematical Reviews (MathSciNet): MR89m:53016
    Zentralblatt MATH: 0672.49028
    Digital Object Identifier: doi:10.2307/2322896
    JSTOR: links.jstor.org
  • [M2] F. Morgan, Calibrations and new singularities in area-minimizing surfaces: A survey, Variational Methods (Paris, 1988) eds. H. Berestycki, J-M. Coron, and I. Ekeland, Progress in Nonlinear Differential Equations and their Applications, vol. 4, Birkhauser, Boston, 1990, pp. 329–342.
    Mathematical Reviews (MathSciNet): MR93i:53009
    Zentralblatt MATH: 0721.53058
  • [M3] F. Morgan, Geometric Measure Theory: A Beginner's Guide, Academic Press, Boston, 1988, second edition, 1995.
    Mathematical Reviews (MathSciNet): MR89f:49036
    Zentralblatt MATH: 0671.49043
  • [M4] F. Morgan, Least-volume representatives of homology classes in $G(2,\mathbf C^4)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 127–135.
    Mathematical Reviews (MathSciNet): MR90c:57030
    Zentralblatt MATH: 0679.53060
  • [M5] F. Morgan, The exterior algebra $\Lambda\sp k\bf R\sp n$ and area minimization, Linear Algebra Appl. 66 (1985), 1–28.
    Mathematical Reviews (MathSciNet): MR86i:53036
    Zentralblatt MATH: 0585.49029
    Digital Object Identifier: doi:10.1016/0024-3795(85)90123-5
  • [P] L.-H. Pan, Existence and uniqueness of volume-minimizing cycles in Grassmann manifolds, Ph.D. thesis, University of Pennsylvania, 1992.
  • [Pe] S. Pedersen, Volumes of vector fields on spheres, Trans. Amer. Math. Soc., to appear.
    Mathematical Reviews (MathSciNet): MR1079056
    Zentralblatt MATH: 0771.53023
    Digital Object Identifier: doi:10.2307/2154338
    JSTOR: links.jstor.org
  • [Wi] W. Wirtinger, Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung, Monatsh. Math. Phys. 44 (1936), 343–365.
    Zentralblatt MATH: 0015.07602
  • [Wo] J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967.
    Mathematical Reviews (MathSciNet): MR36:829
    Zentralblatt MATH: 0162.53304

Duke University Press

Duke Mathematical Journal

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more