On the global regularity of wave maps in the critical Sobolev norm (Q2747848)

The authors extend the recent result by \textit{T. Tao} [Int. Math. Res. Not. 2001, 299-328 (2001; Zbl 0983.35080)] on wave maps \(\varphi: R{n+1}\to (N,h)\) from Minkowski space \(R{n+1}, n\geq 5\) to a tangent Riemannian manifold \((N,h)\) which possesses a ``bounded parallelizable'' structure (the class of such manifolds includes all compact manifolds, Lie groups, homogeneous and hyperbolic spaces.) They prove that if the initial data \((\varphi,\psi)\) of the wave map belong to the Sobolev space \(H^s\) for some \(s>n/2\) and its \(H^{n/2}\)-norm is sufficiently small then the family of wave maps subject to the initial value problem NEWLINE\[NEWLINE \varphi(0)=\varphi, \qquad \partial_t\varphi(0)=\psi NEWLINE\]NEWLINE is globally defined and continuous in the \(H^s\)-norm.