2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122873In this paper we establish the equivalence of solutions between Schrödinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.19 pages, submittedAnalysis of PDEsDifferential Geometrytext