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

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

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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


  • [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.
  • [B] M. Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4) 5 (1972), 1–44.
  • [BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958), 458–538.
  • [B1] K. A. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1 (1991), no. 4, 329–338.
  • [B2] K. A. Brakke, Soap films and covering spaces, preprint, 1993.
  • [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.
  • [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.
  • [DGGW2] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, Conformal isospectral deformations, Indiana Univ. Math. J. 41 (1992), no. 1, 99–107.
  • [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.
  • [DGGW4] D. DeTurck, H. Gluck, C. Gordon, and D. Webb, The inaudible geometry of nilmanifolds, Invent. Math. 111 (1993), no. 2, 271–284.
  • [F1] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969.
  • [F2] H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351–407.
  • [F3] H. Federer, Some theorems on integral currents, Trans. Amer. Math. Soc. 117 (1965), 43–67.
  • [GMZ] H. Gluck, F. Morgan, and W. Ziller, Calibrated geometries in Grassmann manifolds, Comment. Math. Helv. 64 (1989), no. 2, 256–268.
  • [GW] H. Gluck and F. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983), no. 1, 107–132.
  • [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.
  • [GWZ1] H. Gluck, F. Warner, and W. Ziller, The geometry of the Hopf fibrations, Enseign. Math. (2) 32 (1986), no. 3-4, 173–198.
  • [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.
  • [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.
  • [GHV] W. Greub, S. Halperin, and R. Vanstone, Connections, Curvature, and Cohomology, Academic Press, New York, 1976.
  • [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.
  • [H2] R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Boston, 1990.
  • [HL] R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157.
  • [L1] G. Lawlor, A sufficient criterion for a cone to be area-minimizing, Mem. Amer. Math. Soc. 91 (1991), no. 446, vi+111.
  • [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.
  • [M1] F. Morgan, Area-minimizing surfaces, faces of Grassmannians, and calibrations, Amer. Math. Monthly 95 (1988), no. 9, 813–822.
  • [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.
  • [M3] F. Morgan, Geometric Measure Theory: A Beginner's Guide, Academic Press, Boston, 1988, second edition, 1995.
  • [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.
  • [M5] F. Morgan, The exterior algebra $\Lambda\sp k\bf R\sp n$ and area minimization, Linear Algebra Appl. 66 (1985), 1–28.
  • [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.
  • [Wi] W. Wirtinger, Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung, Monatsh. Math. Phys. 44 (1936), 343–365.
  • [Wo] J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967.