On the stability of stationary wave maps (Q1357452)

A wave map is a map \(U\) from a Lorentzian manifold \((M^{m+1},g)\) into a Riemannian manifold \((N^n,h)\) which is a critical point of the Lagrangian \[ {\mathcal L} (U)= {\textstyle {1\over 2}} \int_MTr_gU^*h ={\textstyle {1\over 2}} \int_M|DU|^2 ={\textstyle {1\over 2}} \int_M g^{\alpha\beta} h_{ab} D_\alpha U^a D_\beta U^b. \] In the article under review the case \(M=S^2 \times\mathbb{R}\) and \(N=S^2\) is considered. The existence of stationary maps generated by the symmetry groups and their stability under equivariant perturbations are proved. As consequence of this stability result it is obtained the existence of a large set of initial data, with no degree or energy restrictions, for which the Cauchy problem is globally well posed.