Harmonic Maps, Rigidity, and Hodge Theory

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 $$E(f) = {\smallint _M}{\left| {df} \right|^2}$$ E ( f ) = ∫ M | d f | 2 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.