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

Abstract

In this article we investigate a type of totally geodesic map which has its image being a geodesic in an anisotropic Riemannian manifold. We consider its nonlinear stability among the family of wave maps. We first establish the factorization property and then formulate the stability problem into a PDE system in a specially constructed chart of geodesic normal coordinates. With a generalization of the hyperboloidal foliation, we establish the global existence result associate to small initial data for this PDE system, which leads to the geometric stability.

72 Pages. References are updated. A figure is added. Some concrete examples are available. All comments and suggestions are warmly welcome