Wave maps into the sphere

In this note we discuss some geometric analogs of the classical harmonic functions on Rn and their associated evolutions. Harmonic functions are ubiquitous in mathematics, with applications arising in complex analysis, potential theory, electrostatics, and heat conduction. A harmonic function u in R solves the Laplace equation ∆u = 0 in a domain Ω, where ∆ = ∑n j=1 ∂ 2 xj . The well-known Dirichlet principle says that harmonic functions are critical points of the Dirichlet energy 1 2 ∫ Ω |∇u| 2. The associated evolutions are the heat equation ∂tu − ∆u = 0 and the wave equation ∂2 t u − ∆u = 0. The geometric analogs we will be discussing (which also have applications in physics) are harmonic maps: functions u : R → M , where M is a Riemannian manifold. Introduced in the early 1960s by Eells and Sampson [15], they are also critical points of the Dirichlet energy. The resulting Euler–Lagrange equation is a nonlinear PDE, because of the nonlinear constraint that u(x) ∈M . When M = S2 with the round metric, the equation becomes ∆u = −|∇u|2u. The point of introducing harmonic maps was to use them as a tool to study the geometric and topological properties of the manifold M . For instance, Eells and Sampson showed, using the associated heat flow (the harmonic map heat flow), that smooth functions from R to M can be deformed (under certain geometric conditions on M) into harmonic maps. This work inspired Hamilton [20] to introduce his Ricci flow, which eventually led to the proof of the Poincaré conjecture by Perelman [28]. The study of the singularities of these flows led to the notion of bubbling in singularity formation. It turns out that the bubbling phenomenon is universal, and is analogous to the soliton resolution which we will be considering later. We now turn to wave maps, the wave flow associated with harmonic maps. The topic is vast, and I will concentrate only on aspects close to my interests. There are several ways to define wave maps u : R×R →M . A formal one is to consider the Lagrangian