Dissertation defense titled Uniqueness of Half-Wave Maps into S2 and H2 in dimensions d ≥ 3. The half-wave maps (HWM) equation, ∂tu = u ∧ (−Δ)1/2 u, is a geometric partial differential equation arising from the continuum limit of Calogero-Moser spin systems. Its solutions map Minkowski spacetime into a target manifold M, typically the sphere S2 or the hyperbolic plane
H2. A complete well-posedness theory for this equation, which requires existence and uniqueness of solutions, remains an open problem.
This thesis contributes to the uniqueness theory for the HWM equation in dimensions d ≥ 3. First, we establish a uniqueness result for solutions mapping into the sphere S2 assuming a suitable mixed space–time control associated to the Strichartz estimates for the wave equation in d ≥ 3. The proof leverages the reformulation of the HWM equation into a wave-type equation, followed by a Grönwall argument that relies on fractional Leibniz rules and commutator estimates.
Second, I extend this uniqueness proof to any compact subset of hyperbolic space H2, which introduces a significant geometric complication. To leverage the geometric properties that proved successful in the S2 case, the energy is naturally defined using the Lorentzian inner product. However, this inner product is indefinite in the ambient space. To circumvent this, we employ the Nash
embedding theorem to isometrically embed H2 into some Euclidean space Rm. This allows us to define a non-negative energy with the Euclidean inner product of Rm for the difference of two embedded solutions. A refined Grönwall argument within this framework then proves uniqueness under the same assumptions as before.
This work advances the well-posedness theory for the HWM equation in both S2 and H2 and provides a framework for addressing uniqueness of HWM in other geometric settings.

Advisor: Dr. Armin Schikorra