Wave Maps

is the natural combination of these equations, describing the free motion of an ndimensional surface in a non-Euclidean space; for instance, the motion of a string that is constrained to lie on a sphere would be given (at least to first approximation) as a wave map. As such, wave maps are one of the fundamental equations used to describe geometric motion, although they are not as well understood (and have fewer applications to geometry at present) than other geometric flows such as the Ricci flow or mean curvature flow. One contrast between the wave maps equations and these other flows is that the wave maps equation is time-reversible and non-dissipative; a surface never loses or gains any “energy” under this equation. In contrast, the Ricci and mean curvature flows are dissipative (or parabolic), which means that as time progresses, the surfaces should become smoother and less energetic (although singularities are still possible, and of tremendous importance to geometry). A major difficulty in understanding these equations is that they are non-linear; this may not be so apparent in the formulation (2) but is hidden in the notation for the covariant derivatives ∇t, ∇xj , which we will come to later. Wave maps are also closely related to harmonic maps, which have played an important role in geometry, topology, and even integrable systems; it is thus conceivable that