An Introduction to the Theory of Wave Maps and Related Geometric Problems

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]. …”

Large data global regularity for the classical equivariant Skyrme model

“…The construction of the auxiliary function Φ and further reductions. The proof of Theorem 2.2 is somewhat indirect, in the sense that we argue for (10) by constructing a new function Φ, which satisfies an equation that is easier to study than the one for v (i.e., (7)). The first step in this construction aims to eliminate the derivative terms on the right-hand side of (7) and, for that purpose, we take…”

“…As the value of α does not play an important role in our arguments other than being positive, from here on in, we set it to α = 1 in order to simplify the exposition. For a more comprehensive discussion of the physical descriptions and motivations for both the Gell-Mann-Lévy and Skyrme models, we refer the reader to our recent monograph [7] and references therein. Our focus in this article is on the Euler-Lagrange equations associated to the equivariant case of the Skyrme model; i.e., we work with formal critical points for (1) under the ansatz U (t, r, ω) = (u(t, r), ω), where g = −dt 2 + dr 2 + r 2 dω 2 , h = du 2 + sin 2 u dω 2 , are the previous metrics written in polar form.…”

“…A parallel literature exists on other Skyrme-like theories, like the Faddeev and Adkins-Nappi models, for which we ask the interested reader to consult, yet again, our book [7].…”

Sharp Global Regularity for the 2 + 1-Dimensional Equivariant Faddeev Model

“…where the round metric on S 3 is h = du 2 + sin 2 u dn 2 , 0 ≤ u ≤ π, n ∈ S 2 , and sets α = 1, then the Skyrme action (3) reduces to the Faddeev one (1). We ask the interested reader to consult our monograph [10] and references therein for a more comprehensive view on the physical descriptions and motivations for both models.…”