Global Existence of Small Equivariant Wave Maps on Rotationally Symmetric Manifolds

Abstract

Abstract. We introduce a class of rotationally invariant manifolds, which we call admissible, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity H n 2 . The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces H n for n ≥ 3.

Local in Time Strichartz Estimates for the Dirac Equation on Spherically Symmetric Spaces

“…[5]) and dispersive estimates are well known for them; there is thus the chance to adapt these results to the present setting, and this will be the object of future investigations. We mention that a similar point of view has been developed in [17] in the different context of the study of equivariant wave maps.…”

Global existence of Dirac-wave maps with curvature term on expanding spacetimes

“…Here, we want to mention the works of Choquet-Bruhat [9, 10] for wave maps on Robertson–Walker spacetimes and several recent articles that consider wave maps on non-flat backgrounds [11, 14, 18, 20]. To obtain an overview on the current status of research on the wave map equation we refer to the recent book [12].…”

Global Strichartz estimates for the Dirac equation on symmetric spaces

“…but as a matter of fact this will be implied by our forthcoming assumptions (A2), and therefore we do not strive to optimize on this condition, as done in [10].…”

“…We now introduce weighted spinors, the main goal being transforming the system (2.3) into a system of wave equations on R n perturbed by a radial, electric potential, in order to exploit the existing theory to obtain dispersive estimates. This strategy has been already employed in [2,10] in different contexts (the Schrödinger equation on spherically symmetric manifolds and equivariant wave maps respectively) and in the predecessor of this paper, [5], to deal with the local-in-time case. Take σ : R + → R + such that for all r > 0,…”