Abstract
Let Ω ⊂ Rm be an open set, Nn a Riemannian manifold, X a collection of vector fields on Ω, and f a smooth map from Ω into Nn. We call f a subelliptic harmonic map if it is a critical point of the energy functional with respect to X. In this paper, we calculate the first and the second variations of the energy functional, and use them to prove the partial uniqueness of a subelliptic harmonic map under the condition that Nn has the non-positive curvature. Then, we utilize the maximum principle for subelliptic PDEs to verify the global uniqueness of a subelliptic harmonic map under some other conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bony, J. M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier 19 (1969), 227–304.
Eells, J. and Sampson, J., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 85 (1964), 109–160.
Hildebrandt, S.: Harmonic Mappings of Riemannian Manifolds, Lecture Notes in Math. 1161, Springer-Verlag, Berlin, 1984, pp. 1–117.
Jost, J.: Riemannian Geometry, Geometric Analysis, Springer-Verlag, Berlin, 1998.
Jost, J. and Xu, C. J.: Subelliptic harmonic maps, Trans. AMS 350 (1998), 4633–4649.
Xu, C. J. and Zuily, C.: Higher interior regularity for quasilinear subelliptic systems, Calcul. Var. PDE 5 (1997), 323–343.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhou, ZR. Uniqueness of Subelliptic Harmonic Maps. Annals of Global Analysis and Geometry 17, 581–594 (1999). https://doi.org/10.1023/A:1006620325479
Issue Date:
DOI: https://doi.org/10.1023/A:1006620325479