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Here dσ=Πα=0dxα denotes the standard volume element on R n+1. In local coordinates, one finds the equation (1.1) u + Γjk∂αu ∂u =0, = ∂α∂ We shall refer to this as the local coordinate formulation of the wave maps condition. Another possibility is to isometrically embed the target into a Euclidean space M ↪→ Rk, and then work in terms of the ambient coordinates. This is seen to lead to the condition (1.2) u ⊥ TMu which leads to (1.3) u + S jk(u)∂αu ∂u =0 in terms of the ambient coordinates, where Si jk is the 2nd fundamental form of the embedding M ↪→ Rk. 1Throughout, the Einstein summation convention is in force. c ©2008 International Press

Global regularity and singularity development for wave maps