Schrödinger flow on Hermitian locally symmetric spaces
Given a map \(u_0\) from a Riemannian manifold \(M\) into a complete Kähler manifold \(N\), the flow \(u(x,t)\) defined by the Cauchy problem \(\partial_t u = J(u)\tau(u)\) with the initial condition \(u(x,0)=u_0\) is called the Schrödinger flow, where \(\tau(u)\) denotes the tension field of \(u\). This flow has been investigated from various points of view for many years. The authors in this paper consider a slightly generalized tension field by modifying the energy functional by a coupling function and thus the flow is called the inhomogeneous Schrödinger flow. The main theorem asserts the global existence of the inhomogeneous Schrödinger flow for a smooth coupling function with certain general conditions when \(N\) is a complete Hermitian locally symmetric space and \(M\) is the real line or the \(1\)-dimensional sphere. This result generalizes the global existence result shown by \textit{C. L. Terng} and \textit{K. Uhlenbeck} [Surv. Diff. Geom. IV, Int. Press, Boston MA, 315-402 (1998; Zbl 0935.35163)].
scientific article; zbMATH DE number 1893860