Nonlinear stability of the rank-one totally geodesic wave maps in non-isotropic manifolds

In this presentation I will talk about some recent progress on the global stability problem of totally geodesic wave maps defined in $\mathbb{R}^{1+2}$. We firstly prove the factorization property of the rank-one totally geodesic maps, which was taken as an a priori hypothesis in previous works. Then we reformulate the problem when the target space is an non-isotropic Riemannian manifold. With some symmetry along the target geodesic, we have found that the evolution system of the perturbation still enjoys sufficiently nice structure such that the global stability can be established via a global analysis on a type of wave-like systems defined in $\mathbb{R}^{1+2}$. For our purpose, two additional technical tools are developed: the conformal energy estimate on the glued hyper-surfaces and a variation of the algebraic normal form transform. This is a joint work with S.-H. Duan and W.-D. Zhang.