Abstract
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M n. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section σ: M → B is the mass of the image σ(M) as a rectifiable n-current in B.
Theorem 1. For any homology class of sections of B, there is a mass-minimizing rectifiable current T representing that homology class which is the graph of a C1 section on an open dense subset of M.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almgren, F.J., 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, 327–328 (1983)
Bombieri, E.: Regularity theory for almost minimal currents. Arch. Rat. Mech. Anal. 78, 99–130 (1982)
Coventry, A.: Research reports (Mathematics), number CMA-MRR 45–98, Australian National University Publications (1998)
Evans, L.C,. Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Fanghua, L., Xiaoping, Y.: Geometric Measure Theory, An Introduction, International Press, Boston, (2002)
Federer, H.: Geometric Measure Theory, Springer-Verlag, Berlin, (1969)
Johnson, D.L.: Kähler submersions and holomorphic connections. J. Differential Geom. 15, 71–79 (1980)
Johnson, D.L., Smith, P.: Regularity of volume-minimizing graphs. Indiana Univ. Math. J. 44, 45–85 (1995)
Johnson, D.L., Smith, P.: Regularity of mass-minimizing one-dimensional foliations. In: Analysis and Geometry on Foliated Manifolds, Proceedings of the VII International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98.
Moore, J.D., Schlafly, R.: On equivariant isometric embeddings. Math. Z. 173, 119–133 (1980)
Morgan, F.: Geometric Measure Theory, A Beginner's Guide, 2nd edn., 1995, Academic Press (1988)
Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer-Verlag, Berlin, (1966)
Roubíček, T.: Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, (1999)
Sasaki, T.: On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J. 10, 338–354 (1958)
Serrin, J.: On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101, 139–167 (1961)
Struwe, M.: Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag Ergeb. Math. Grenzgeb. 34, (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000): 49F20, 49F22, 49F10, 58A25, 53C42, 53C65.
Rights and permissions
About this article
Cite this article
Johnson, D.L., Smith, P. Partial regularity of mass-minimizing rectifiable sections. Ann Glob Anal Geom 30, 239–287 (2006). https://doi.org/10.1007/s10455-006-9025-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-006-9025-9