The adiabatic limit of wave map flow on a two-torus

J. M. Speight
2015 Transactions of the American Mathematical Society  
The S 2 valued wave map flow on a Lorentzian domain R × Σ, where Σ is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Σ → S 2 is considered, in the limit of small initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the L 2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward. * It
more » ... ollows (from Noether's Theorem) that they conserve the total energy T + E. A rather general argument of Lichnerowicz [12] shows that for any map φ : Σ → S 2 of topological degree n ∈ Z (subject to suitable boundary conditions, if Σ is noncompact), E ≥ 4π|n|, with equality if and only is φ is ± holomorphic. So holomorphic maps, if they exist, minimize potential energy in their homotopy class. Let us denote by M n the moduli space of degree n holomorphic maps Σ → S 2 . Consider a wave map L 2 , one expects that φ(t) will stay close to M n , on which E attains its minimum value, for as long as the solution persists. This led Ward to suggest [29] , in the specific case Σ = R 2 , that such wave maps should be well approximated by the dynamical system with action S, but with φ(t) constrained to M n for all time. Since E is constant on M n , this constrained system is equivalent to geodesic motion on M n with respect to the L 2 metric (obtained by restricting the quadratic form T to T M n ). A similar approximation had previously been proposed by Manton [15] for low energy monopole dynamics, and the geodesic approximation is now a standard technique in the study of the dynamics of topological solitons [16] . Geodesic motion on M 2 (for Σ = R 2 ) was studied in detail in [10] . There is a technical problem: the L 2 metric is only well-defined on the leaves of a foliation of M n and one must impose by hand that φ(t) remains on a single leaf. This turns out to be ill-justified (it precludes singularity formation for n = 1, for example, in contradiction of [8, 18] ). This technical deficiency is removed if we choose Σ to be a compact Riemann surface. Here geodesic motion in M n is globally well-defined, if incomplete [20] , and the L 2 geometry of M n is quite well understood, at least for some choices of Σ and n [14, 22, 23, 24] . The question remains: is geodesic motion in M n really a good approximation to wave map flow in the adiabatic (low velocity) limit? The purpose of this paper is to prove that it is, for times of order (initial velocity) −1 at least in the case where Σ is any flat two-torus. More precisely, we will prove: Theorem 1.1 (Main Theorem). Let M n denote the moduli space of degree n ≥ 2 holomorphic maps from a flat two-torus Σ to S 2 . For fixed φ 0 ∈ M n and φ 1 ∈ T φ 0 M n consider the one parameter family of initial value problems for the wave map equation with φ(0) = φ 0 , φ t (0) = εφ 1 , parametrized by ε > 0. There exist constants τ * > 0 and ε * > 0, depending only on the initial data, such that for all ε ∈ (0, ε * ], the problem has a unique solution for t ∈ [0, τ * /ε]. Furthermore, the time re-scaled solution φ ε : [0, τ * ] × Σ → S 2 , φ ε (τ, p) = φ(τ /ε, p) converges uniformly in C 1 to ψ : [0, τ * ] × Σ → S 2 , the geodesic in M n with the same initial data, as ε → 0.
doi:10.1090/tran/6538 fatcat:3gss7btwwzgazogbrkfmsoglsu