Planar minimal surfaces with polynomial growth in the $\mathrm{S}\mathrm{p}(4,\mathbb{R})$-symmetric space

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{S}\mathrm{p}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the “Hitchin component” of wild $\mathrm{S}\mathrm{p}(4,\mathbb{R})$-Higgs bundles over ${\mathbb{C}\mathbb{P}}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in ${\mathbb{H}}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of ${\mathbb{R}}^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in ${\mathbb{H}}^{2,2}$ associated to $\mathrm{S}\mathrm{p}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials.