Existence results for the nonlinear Hodge minimal surface energy

  • Daniel AgressEmail author


Given a compact Riemannian manifold \((M^n,g)\) and a fixed cohomology class, \([\alpha ^*] \in H^k(M)\), we consider the existence of a minimizer \(\alpha \in [\alpha ^*]\) of the generalized minimal surface energy \(\int _M \sqrt{1+|\alpha |^2} dV_g\). When \(k = 1\), we prove the existence of unique minimizers for every cohomology class \([\alpha ^*]\). Next, when \(k > 1\), we construct examples of singular solutions for finite cohomology class \([\alpha ^*] \in H^k(S^k \times S^k,g)\), where g is conformal to the standard metric on \(S^k \times S^k\). Additionally, we show that when \(k=2\), these singular solutions are also solutions to the Born Infeld equation.

Mathematics Subject Classification

58J05 35J50 49Q05 



The author would like to thank his advisors Patrick Guidotti and Jeffrey Streets, as well as Richard Schoen for their insight and advice.


  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, vol. 254. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  2. 2.
    Bombieri, E., De Giorgi, E., Miranda, M.: Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Ration. Mech. Anal. 32(4), 255–267 (1969)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Born, M., Infeld, L.: Foundations of the new field theory. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character 144(852), 425–451 (1934)CrossRefGoogle Scholar
  4. 4.
    Braides, A.: Gamma-Convergence for Beginners, vol. 22. Clarendon Press, Oxford (2002)CrossRefGoogle Scholar
  5. 5.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80. Birkhauser, Basel (1984)CrossRefGoogle Scholar
  6. 6.
    Sibner, L.: An existence theorem for a non-regular variational problem. Manuscr. Math. 43(1), 45–72 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sibner, L., Sibner, R.: A non-linear hodge-de rham theorem. Acta Math. 125(1), 57–73 (1970)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sibner, L., Sibner, R.: Nonlinear hodge theory: applications. Adv. Math. 31(1), 1–15 (1979)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sibner, L., Sibner, R.: Transonic flow on an axially symmetric torus. J. Math. Anal. Appl. 72(1), 362–382 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematics and its Applications, Centre for Mathematical Analysis, vol. 3, The Australian National University, Canberra (1984)Google Scholar
  11. 11.
    Spruck, J.: Interior gradient estimates and existence theorems for constant mean curvature graphs in \({M}^n \times \mathbb{R}\). Pure Appl. Math. Q. 3(3), 785–800 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(1), 219–240 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California IrvineIrvineUSA

Personalised recommendations