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

Andrea Tamburelli, Michael Wolf

TL;DR

The paper studies planar minimal surfaces in the symmetric space $ ext{Sp}(4,\mathbb{R})/ ext{U}(2)$ that arise from Higgs bundles with polynomial quartic data, establishing a bridge between polynomial quartic differentials on $\mathbb{C}$ and geometric boundary data in Ein^{1,2}. It constructs a conformal harmonic map $f:\mathbb{C} o ext{Sp}(4,\mathbb{R})/ ext{U}(2)$ from a meromorphic irregular Higgs bundle, solves Hitchin’s equations with a diagonal metric, and analyzes the asymptotics of the associated minimal surface. The authors show that these planar maximal surfaces are asymptotic to a discrete collection of flats and exhibit a Stokes-type transition across half-planes, with precise error control. They also realize a convex immersion into the Grassmannian Gr_2(𝔈_ℝ), relate it to maximal surfaces in $ ext{H}^{2,2}$ via the $ ext{Sp}(4,\mathbb{R}) o ext{SO}_0(2,3)$ isomorphism, and describe a boundary correspondence sending polynomial quartics to future-directed negative light-like polygons in Ein^{1,2}. Finally, they prove that the moduli space of polynomial quartic differentials of degree $n$ is homeomorphic to a connected component of the moduli space of such light-like polygons, and establish localization results along rays that connect the analytic PDE analysis to geometric boundary data.

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.

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

TL;DR

The paper studies planar minimal surfaces in the symmetric space $ ext{Sp}(4,\mathbb{R})/ ext{U}(2)$ that arise from Higgs bundles with polynomial quartic data, establishing a bridge between polynomial quartic differentials on $\mathbb{C}$ and geometric boundary data in Ein^{1,2}. It constructs a conformal harmonic map $f:\mathbb{C} o ext{Sp}(4,\mathbb{R})/ ext{U}(2)$ from a meromorphic irregular Higgs bundle, solves Hitchin’s equations with a diagonal metric, and analyzes the asymptotics of the associated minimal surface. The authors show that these planar maximal surfaces are asymptotic to a discrete collection of flats and exhibit a Stokes-type transition across half-planes, with precise error control. They also realize a convex immersion into the Grassmannian Gr_2(𝔈_ℝ), relate it to maximal surfaces in $ ext{H}^{2,2}$ via the $ ext{Sp}(4,\mathbb{R}) o ext{SO}_0(2,3)$ isomorphism, and describe a boundary correspondence sending polynomial quartics to future-directed negative light-like polygons in Ein^{1,2}. Finally, they prove that the moduli space of polynomial quartic differentials of degree $n$ is homeomorphic to a connected component of the moduli space of such light-like polygons, and establish localization results along rays that connect the analytic PDE analysis to geometric boundary data.

Abstract

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

Paper Structure

This paper contains 26 sections, 51 theorems, 330 equations, 2 tables.

Table of Contents

  1. Background material
  2. Lie theory for $\mathrm{Sp}(4,\mathbb{R})$
  3. $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles
  4. Planar minimal surfaces with polynomial growth
  5. Moduli space of polynomial quartic differentials
  6. Existence
  7. Geometry of the minimal surface
  8. Construction of the minimal surface
  9. The case of constant quartic differential
  10. The general case
  11. Standard half-planes and rays
  12. Comparing $\psi_{0}$ and $\psi$
  13. Asymptotic behaviour of the minimal surface
  14. The induced metric on the minimal surface
  15. Estimates of the error term
  16. ...and 11 more sections

Key Result

Theorem 2

Assume that $q_{4}$ is a polynomial holomorphic quartic differential of degree $n$. Then there exists a conformal harmonic map $f: \mathbb{C} \rightarrow \mathrm{Sp}(4,\mathbb{R})/\mathrm{U}(2)$ such that $q_{4}=\mathrm{tr}((\partial f)^{4})$. Moreover, the associated minimal surface $f(\mathbb{C})$

Theorems & Definitions (108)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Theorem 2.1
  • Definition 2.2
  • ...and 98 more