Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 068, 8 pages      arXiv:0903.1018      https://doi.org/10.3842/SIGMA.2009.068
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Boundaries of Graphs of Harmonic Functions

Daniel Fox
Mathematics Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, UK

Received March 06, 2009, in final form June 16, 2009; Published online July 06, 2009

Abstract
Harmonic functions u: Rn → Rm are equivalent to integral manifolds of an exterior differential system with independence condition (M,I,ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g: Sn–1 → M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary g(Sn–1). The proof uses standard linear elliptic theory to produce an integral manifold G: Dn → M and the completeness of the space of conservation laws to show that this candidate has g(Sn–1) as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in Cm in the local case.

Key words: exterior differential systems; integrable systems; conservation laws; moment conditions.

pdf (196 kb)   ps (147 kb)   tex (10 kb)

References

  1. Bochner S., Analytic and meromorphic continuation by means of Green's formula, Ann. of Math. (2) 44 (1943), 652-673.
  2. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Springer-Verlag, New York, 1991.
  3. Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (1995), 507-596.
  4. Gilbarg D., Trudinger N.S., Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001.
  5. Harvey F.R., Lawson H.B. Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), 223-290.
  6. Harvey F.R., Lawson H.B. Jr., On boundaries of complex analytic varieties. II, Ann. Math. (2) 106 (1977), 213-238.
  7. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 2003.
  8. Wermer J., The hull of a curve in Cn, Ann. of Math. (2) 68 (1958), 550-561.

Previous article   Next article   Contents of Volume 5 (2009)