The Cauchy problem for wave maps on a curved background

Abstract

Abstract. We consider the Cauchy problem for wave maps u : R × M → N , for Riemannian manifolds (M, g) and (N, h). We prove global existence and uniqueness for initial data,, where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe in [31] for proving global existence and uniqueness of small data wave maps u : R × R d → N in the critical norm, for d ≥ 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru in [24].

The Cauchy Problem for Wave Maps on Hyperbolic Space in Dimensions $d\geq 4$

“…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]. Also, we note the recent work of D'Ancona, Zhang [18] who proved global existence for small equivariant wave maps on rotationally symmetric spacetimes for d ≥ 3.…”

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 Existence of Small Equivariant Wave Maps on Rotationally Symmetric Manifolds

“…with µ(r) odd, smooth, with µ ′ (0) = 0, satisfying the following assumptions: (5.9) Then Proposition 5.5 implies immediately that the metric h ǫ is admissible if ǫ is sufficiently small. Note that in dimension n = 4 this result is essentially a corollary of Theorem 1.1 in [27]. In that paper the global existence of small wave maps is proved on four dimensional, asymptotically flat manifolds without symmetry assumptions.…”