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Global existence of Dirac-wave maps with curvature term on expanding spacetimes. (English) Zbl 1415.58017

Wave maps are among the fundamental variational problems in differential geometry. They are defined as critical points of the Dirichlet energy for a map between two manifolds, where one assumes that the domain manifold is Lorentzian and the target manifold is Riemannian. There are less articles that consider the case of a domain being a non-flat globally hyperbolic manifold. The authors prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, they also prove a global existence result for wave maps under similar assumptions.

MSC:

58J45 Hyperbolic equations on manifolds
53C27 Spin and Spin\({}^c\) geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
35L71 Second-order semilinear hyperbolic equations
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