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

Abstract

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the "Hitchin component" of wild $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles over ${\mathbb{CP}}^1$ 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 $\mathrm{Sp}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials.

68 pages

References in corpus (2)

• G2 geometry and integrable systems
• From $S^1$-fixed points to $\mathcal{W}$-algebra representations

Cited by in corpus (4)

• Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces
• Isolated singularities of Toda equations and cyclic Higgs bundles
• Length spectrum compactification of the ${\mathrm{SO}}_0(2,3)$-Hitchin component
• Boundary of the Gothen components