Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the (4,)-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild (4,)-Higgs bundles over ¹ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in 2,2. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of ⁴. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in 2,2 associated to (4,)-Hitchin representations along rays of holomorphic quartic differentials.