2016
DOI: 10.3934/dcds.2016.36.3857
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Stability of stationary wave maps from a curved background to a sphere

Abstract: We study time and space equivariant wave maps from M × R → S 2 , where M is diffeomorphic to a two dimensional sphere and admits an action of SO(2) by isometries. We assume that metric on M can be written as dr 2 + f 2 (r)dθ 2 away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared … Show more

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Cited by 3 publications
(2 citation statements)
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“…Shatah, Tahvildar-Zadeh showed the existence and orbital stability of equivariant timeperiodic wave maps, R × S 2 → S 2 . This was extended by the third author [56] to allow for maps R × Σ → S 2 where Σ is diffeomorphic to S 2 and admits an SO(2) action. The first author established a critical small data global theory for wave maps on small asymptotically flat perturbations of R 4 [39] using the linear estimates of Metcalfe, Tataru [48].…”
Section: 2mentioning
confidence: 99%
“…Shatah, Tahvildar-Zadeh showed the existence and orbital stability of equivariant timeperiodic wave maps, R × S 2 → S 2 . This was extended by the third author [56] to allow for maps R × Σ → S 2 where Σ is diffeomorphic to S 2 and admits an SO(2) action. The first author established a critical small data global theory for wave maps on small asymptotically flat perturbations of R 4 [39] using the linear estimates of Metcalfe, Tataru [48].…”
Section: 2mentioning
confidence: 99%
“…We also mention the works of [32,35,36] where we proved asymptotic stability of harmonic maps under the wave map between hyperbolic planes. And for wave maps on product spaces of spheres and Euclidean spaces, Shatah, Tahvildar-Zadeh [50] and Shahshahani [48] studied orbital stability of stationary solutions.…”
Section: Introductionmentioning
confidence: 99%