Volume-minimizing cycles in Grassmann manifolds

previous :: next

Volume-minimizing cycles in Grassmann manifolds

Herman Gluck, Dana Mackenzie, and Frank Morgan

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

First Page: Show

Primary Subjects: 53C40

Secondary Subjects: 53C65, 57R20, 58E99

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

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077285156
Mathematical Reviews number (MathSciNet): MR1344765
Zentralblatt MATH identifier: 0837.53035
Digital Object Identifier: doi:10.1215/S0012-7094-95-07909-5

back to Table of Contents

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

previous :: next

Duke Mathematical Journal

Turn MathJax Off
What is MathJax?