Global Regularity of Wave Maps from R2+1 to H2. Small Energy

Abstract

Abstract. We demonstrate that Wave Maps with smooth initial data and small energy from R 2+1 to the Lobatchevsky plane stay smooth globally in time. Our method is similar to the one employed in [18]. However, the multilinear estimates required are considerably more involved and present novel technical challenges. In particular, we shall have to work with a modification of the functional analytic framework used in [30], [33], [18].

Geometric renormalization of large energy wave maps

“…In the case of the sphere N = S m−1 , this was achieved for n ≥ 2 in [64], [65] via the microlocal renormalization method; for the more general class of boundedly parallelizable manifolds (which includes the hyperbolic spaces H m ), this was achieved for n ≥ 5 in [27] (by a hybrid of the microlocal renormalization and Coulomb gauge methods) and then for n ≥ 4 in [51], [38] and n ≥ 3 in [31] (by use of the Coulomb gauge). For the hyperbolic plane N = H 2 the n = 2 case was recently treated in [32] (again using the Coulomb gauge), while for manifolds which are uniformly isometrically embeddable in Euclidean space the result was obtained for all n ≥ 2 in [68] (together with more precise well-posedness results). While the result in [68] does not directly cover the hyperbolic spaces H m , which cannot be uniformly isometrically embedded, it may be possible that the argument can be modified to treat this case by first quotienting H m by a discrete group in order to compactify the target, and lifting back to H m at the end of the argument.…”

“…More specifically, one begins to encounter difficulty in large energy in keeping the microlocal gauge change approximately unitary, and in the Coulomb gauge one has problems establishing uniqueness, regularity, and ellipticity of the gauge in the non-abelian case where the target has dimension m > 2 and so the gauge group SO(m) is non-abelian. Also, the microlocal gauge has not yet been successfully applied to manifolds such as hyperbolic space, which are not easy to embed inside Euclidean spaces; meanwhile, the Coulomb gauge introduces some additional (and somewhat artificial) singularities at the spatial frequency origin ξ = 0 (arising from the Hodge decomposition) which cause additional technical complications, especially in two dimensions (see [31], [32] for a demonstration of this phenomenon). Finally, while both these gauges are well suited for establishing regularity of a single solution, the task of establishing a perturbation theory -i.e.…”

“…all derivatives ∇ N x,t (φ, e) decay faster than any polynomial) to some point (φ(∞), e(∞)) ∈ Frame(N) at infinity 5 This then implies that (ψ α , A α ) are Schwartz. To reconstruct (φ, e) from 4 The cancellation inherent in this wedge product operation will be largely unexploited, in contrast to [32], [31], where this type of "Q ij null structure" is needed to compensate for the singularity of ∇ −1 in the Coulomb gauge. We will, however, rely crucially on the negativity of κ to ensure that the heat flow converges properly.…”

“…Combining this equation with the compatibility conditions (11), (13) and the definitions (7), (9) we can thus write the wave map equation as the system (cf. [51], [38], [31], [32])…”

“…(29) Note that we are specifying data at both s = 0 and s = +∞. As we shall explain shortly, this gauge is the analogue of the microlocal gauge in [64], [65], [27], [68] in the differentiated setting, and will be used here in place of the Coulomb gauge that was used in [38], [51], [31], [32], for reasons which will also be discussed shortly.…”