The Cauchy problem about Dirac-wave map from the 2-dimension Minkowski space to a complete Riemannian manifold (Q2454660)

Dirac-wave maps from \(2\)-dimensional Minkowski space \(\mathbb R^{1+1}\) to a Riemannian manifold \(M\) are the critical points of a functional \({\mathcal L}(\phi,\psi)=\int_{\mathbb R^{1+1}}\{| d\phi| ^2+\langle\psi,/{\mathcal D}\psi \rangle\}\), where \(\phi:\mathbb R^{1+1}\to M\) is a mapping, \(\psi\in\Sigma \mathbb R^{1+1} \otimes\phi^{-1}TM\) a section, \(\Sigma \mathbb R^{1+1}\) the spin bundle of \(\mathbb R^{1+1}\), and \({\mathcal D}\) the Dirac operator along \(\phi\). For complete target manifolds \(M\) of bounded Riemannian curvature, it is proven that the Cauchy problem for Dirac-wave maps on \(\mathbb R^{1+1}\) has unique global solutions. This generalizes a theorem from [\textit{X. Han}, Calc. Var. Partial Differ. Equ. 23, 193--204 (2005; Zbl 1071.53028)] where the same is proven for compact targets.