The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

We consider the half-wave maps equation $$\partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u}$$ for $\mathbf{u} : \mathbb{R} \times \mathbb{T} \to \mathbb{S}^2$, where $\mathbb{T}=\mathbb{R}/2 \pi \mathbb{Z}$ is the one-dimensional torus and $\mathbb{S}^2 \subset \mathbb{R}^3$ denotes the unit sphere. By extension from rational initial data, we construct a unique and continuous flow map for data in the critical energy space $H^{1/2}(\mathbb{T}; \mathbb{S}^2)$. Moreover, we show almost periodicity in time of these solutions. For the dense subset of rational initial data, we establish quasi-periodicity in time and a-priori bounds on $\| \mathbf{u}(t) \|_{H^s(\mathbb{T})}$ for any $s >0$. Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of vector-valued holomorphic functions on the unit disk. As a central ingredient, we develop a general {\em stability principle} for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin--Ono equation, Calogero--Sutherland DNLS, and the half-wave-maps equation posed on $\mathbb{T}$. Our results extend to the matrix-valued half-wave maps equation $$\partial_t \mathbf{U} = -\frac{i}{2} [ \mathbf{U}, |D| \mathbf{U} ]$$ with target manifold given by the complex Grassmannians $\mathsf{Gr}_k(\mathbb{C}^d)$, thereby generalizing the special case $\mathbb{S}^2 \cong \mathbb{CP}^1 \cong \mathsf{Gr}_1(\mathbb{C}^2)$. In a companion work, we prove global well-posedness for the half-wave maps equation posed on $\mathbb{R}$ in the scaling-critical energy space $\dot{H}^{1/2}$, by establishing a stability principle for explicit formulae on Hardy spaces in the complex half-plane $\mathbb{C}_+$.