Equivariant Schrödinger map flow on two dimensional hyperbolic space

In this article, we consider the Schrödinger flow of maps from two dimensional hyperbolic space H² to sphere S². First, we prove the local existence and uniqueness of Schrödinger flow for initial data u0 ∈ H³ using an approximation scheme and parallel transport introduced by McGahagan [32]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrödinger flow to that of a system of coupled Schrödinger equations with potentials. Then we prove the global existence of equivariant Schrödinger flow for small initial data u0 ∈ H¹ by Strichartz estimates and perturbation method.