|
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
- Bochner S.,
Analytic and meromorphic continuation by means of Green's formula,
Ann. of Math. (2) 44 (1943), 652-673.
- Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A.,
Exterior differential systems, Springer-Verlag, New York, 1991.
- Bryant R.L., Griffiths P.A.,
Characteristic cohomology of differential systems. I. General theory,
J. Amer. Math. Soc. 8 (1995), 507-596.
- Gilbarg D., Trudinger N.S.,
Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001.
- Harvey F.R., Lawson H.B. Jr.,
On boundaries of complex analytic varieties. I,
Ann. of Math. (2) 102 (1975), 223-290.
- Harvey F.R., Lawson H.B. Jr.,
On boundaries of complex analytic varieties. II,
Ann. Math. (2) 106 (1977), 213-238.
- 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.
- Wermer J.,
The hull of a curve in Cn,
Ann. of Math. (2) 68 (1958), 550-561.
|
|