Abstract
Harmonic maps are nonlinear analogues of harmonic functions or, if one considers their differentials, harmonic 1-forms. As such, one can expect analogues of Hodge-theoretic results about harmonic 1-forms. Harmonic maps arise as critical points for the energy functional on maps between two Riemannian manifolds. If M, N are Riemannian manifolds and f : M → N is a smooth map between them, then the energy is defined by
where df is the differential of f. If f has finite energy, then we can ask whether f is a critical point for E; the corresponding Euler-Lagrange equation in D* df = 0, where D is the exterior derivative operator associated to the natural connection of f*TN and df is regarded as a 1-form on M with values in f*TN. The latter is the harmonic map equation. It is nonlinear analogue of Laplace’s equation.
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© 1995 Birkhäser Verlag, Basel, Switzerland
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Corlette, K. (1995). Harmonic Maps, Rigidity, and Hodge Theory. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_39
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DOI: https://doi.org/10.1007/978-3-0348-9078-6_39
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