Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons

We study the moduli space of complete maximal surfaces in pseudo-hyperbolic space ${\mathbf{H}}^{2,2}$ with polygonal asymptotic boundary a lightlike $(n+4)$-gon in the Einstein universe ${\mathbf{E}\mathbf{i}\mathbf{n}}^{1,2}$, up to the action of ${\mathsf{S}\mathsf{O}}_0(2,3)$. Importantly, this space $M_{n+4}({\mathbf{H}}^{2,2})$ is disconnected for $n\ge 2$. We introduce a \emph{turning invariant} $\mathrm{\Theta }$ that classifies the components in terms of the geometry of the boundary polygons in the Einstein universe ${\mathbf{E}\mathbf{i}\mathbf{n}}^{1,2}$, which is related to \emph{spherical polygons} in ${\mathbb{S}}^2$. We provide a geometric parametrization of the moduli space $M_{n+4}({\mathbf{H}}^{2,2})$ via holomorphic data, namely a pair of polynomials $\alpha$, $\beta$. These polynomials appear in relation to certain components of the complexified second fundamental form of the corresponding surface. We then relate the holomorphic data $\alpha ,\beta$ with the turning invariant $\mathrm{\Theta }$. This leads to characterizations of the \emph{Hitchin component} of polygonal maximal surfaces studied by Tamburelli-Wolf.