The author establishes global well-posedness and regularity for wave maps from \(\mathbb{R}^{2+1}\) to a large class of manifolds \(M\) when the energy is small; more precisely, this claim is established when \(M\) is isometrically embeddable in a Euclidean space in a uniformly smooth and uniformly locally graphable fashion. In particular, all smooth compact manifolds are covered by this result, as well as many classes of non-compact manifolds, with the notable exception of hyperbolic spaces (which have been treated by J. Krieger [Commun. Math. Phys. 250, No. 3, 507–580 (2004; Zbl 1099.58010)]). The case of the sphere had been treated earlier by the reviewer [Commun. Math. Phys. 224, 443–544 (2001; Zbl 1020.35046)]. There are several further results in higher dimensions (which is significantly simpler) which are too numerous to state here.
One notable feature of this work, not present in earlier results, is that the author not only establishes global persistence of regularity (i.e. smooth initial data remains globally smooth), but also establishes continuous dependence on the data in the energy space, assuming that either time is localized, or some supercritical norm stays bounded.
As with earlier work, a key device used is a certain microlocal renormalization of the wave map. In this paper, this renormalization is achieved via a nonlinear version of Littlewood-Paley theory in which a map taking values in \(M\) is smoothly deformed to a constant map; this nonlinear Littlewood-Paley theory is constructed by taking the linear Littlewood-Paley theory in the ambient space and projecting it carefully onto \(M\). A related device is used to compare two different maps by introducing a smooth deformation from one to the other.
Another technical tool employed is the use of a nonlinear Moser-type estimate, showing that certain technical spaces are well-behaved under composition with smooth functions.