On uniqueness for hyperbolic half-wave maps in dimension $d\ge 3$

Abstract

Half-wave maps appear in the physics literature as the continuum limit of Calogero-Moser spin systems. We obtain a uniqueness result for the Half-Wave Maps equation in dimension $d\ge 3$ in the natural energy class with ${\mathbb{H}}^2$ target. In the proof, we differentiate in time to arrive at a wave-type equation and isometrically embed ${\mathbb{H}}^2$ into some ${\mathbb{R}}^m$ using the Nash embedding theorem. Relying on geometric properties of ${\mathbb{H}}^2$, combined with fractional Leibniz rules and commutator estimates, we then use a Grönwall inequality argument to obtain uniqueness.