Harmonic maps

Abstract

A harmonic map between two Riemannian manifolds (M, g) and (N, γ) of dimension n and m respectively is, roughly speaking, a critical point for the Dirichlet integral

$$ \varepsilon (u): = \int_M {\left| {\nabla u} \right|} ^2 d{\kern 1pt} vol_M ,$$

where, for x ∈ M and charts ϕ and ψ at x and u(x) respectively, and ū: = ψ o u o ϕ⁻¹,

$$\left| {\nabla u} \right|^2 (x): = \gamma _{ij} g^{\alpha \beta } D_\alpha \bar u_{\phi \left( x \right)}^i D_\beta \bar u_{\phi \left( x \right)}^j ,$$

with (g ^αβ) = (g _αβ)⁻¹. If M = Ω ⊂ (ℝⁿ and N = ℝⁿ, then harmonic maps are simply maps whose components are harmonic functions. In general the curvature of N introduces an important nonlinearity in the problem.