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

TL;DR

This paper proves a uniqueness result for the half-wave maps equation in dimensions three and higher, using geometric embeddings and energy estimates, which advances understanding of these equations in mathematical physics.

Contribution

It establishes a new uniqueness theorem for half-wave maps in higher dimensions with hyperbolic target, employing geometric and analytical techniques.

Findings

Proved uniqueness in the natural energy class for half-wave maps in $d \, \geq \, 3$.

Utilized Nash embedding and fractional calculus tools in the proof.

Applied Grönwall inequality to conclude the argument.

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 Gr\"{o}nwall inequality argument to obtain uniqueness.