Wave maps in gravitational theory

Abstract

We survey results on wave maps in gravitational theory. In the first part, we review global properties of cosmological spacetimes with two spacelike commutable Killing vector fields in the Einstein-Maxwell-dilaton-axion system, which is arising in the low energy effective superstring theory. It is shown that the dynamical evolution parts of this Einstein-matter equations become a system of wave map equations and global existence and inextendibility for the system are discussed. Asymptotic behavior of constant mean curvature foliations is also analyze. In the second part, we show a global existence theorem of a wave map on black hole background by using Choquet-Bruhat's arguments. Moreover, asymptotic decay property is shown by conformal transformation.