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Unstable minimal surfaces in symmetric spaces of non-compact type

Nathaniel Sagman, Peter Smillie

TL;DR

The paper addresses the existence and uniqueness of $ ho$-invariant minimal surfaces in symmetric spaces arising from Hitchin representations, proving that in rank at least $3$ there exist Hitchin representations with unstable invariant minimal maps, thus contradicting Labourie’s uniqueness conjecture in higher rank. The authors develop a comprehensive framework based on Higgs bundles, Hitchin rays, and index theory: high-energy limits along the $oldsymbol{R}^+$-flow converge to toral/abelian limiting objects (via cameral covers), and a key index inequality $ ext{Ind}(H_R) o ext{Ind}(f)$ ensures instability transfers from the limit to the high-energy maps. They then show that the Hitchin section parametrizes the Hitchin component in a way that yields explicit unstable minimal surfaces for $ ext{PSL}(n,oldsymbol{R})$ with $n\,oldsymbol{ ge}\,4$, thereby proving non-uniqueness of area-minimizing equivariant minimal surfaces and disproving the original Labourie conjecture as well as its generalized form in higher rank. The results illuminate fundamental limitations of extending Teichmüller-type parametrizations to higher Teichmüller spaces and highlight the delicate interplay between Higgs bundle data, energy functionals, and geometric minimality in symmetric spaces.

Abstract

We prove that if $Σ$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation $ρ:π_1(Σ)\to G$ and a $ρ$-equivariant unstable minimal map from the universal cover of $Σ$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\geq 4$, this disproves the Labourie conjecture.

Unstable minimal surfaces in symmetric spaces of non-compact type

TL;DR

The paper addresses the existence and uniqueness of -invariant minimal surfaces in symmetric spaces arising from Hitchin representations, proving that in rank at least there exist Hitchin representations with unstable invariant minimal maps, thus contradicting Labourie’s uniqueness conjecture in higher rank. The authors develop a comprehensive framework based on Higgs bundles, Hitchin rays, and index theory: high-energy limits along the -flow converge to toral/abelian limiting objects (via cameral covers), and a key index inequality ensures instability transfers from the limit to the high-energy maps. They then show that the Hitchin section parametrizes the Hitchin component in a way that yields explicit unstable minimal surfaces for with , thereby proving non-uniqueness of area-minimizing equivariant minimal surfaces and disproving the original Labourie conjecture as well as its generalized form in higher rank. The results illuminate fundamental limitations of extending Teichmüller-type parametrizations to higher Teichmüller spaces and highlight the delicate interplay between Higgs bundle data, energy functionals, and geometric minimality in symmetric spaces.

Abstract

We prove that if is a closed surface of genus at least 3 and is a split real semisimple Lie group of rank at least acting faithfully by isometries on a symmetric space , then there exists a Hitchin representation and a -equivariant unstable minimal map from the universal cover of to . This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking , , this disproves the Labourie conjecture.
Paper Structure (29 sections, 35 theorems, 77 equations)

This paper contains 29 sections, 35 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. The Labourie Conjecture
  3. Main result
  4. The case $\textrm{PSL}(2,\mathbb R)^n$
  5. Hitchin rays and the idea of the proof
  6. The index of minimal maps
  7. Acknowledgements
  8. Preliminaries
  9. Hermitian metrics
  10. Harmonic maps, bundles, and minimal surfaces
  11. Flat Riemannian bundles
  12. Harmonic and minimal bundles
  13. Higgs bundles
  14. Symmetric spaces and $G$-Higgs bundles
  15. Symmetric spaces of non-compact type
  16. ...and 14 more sections

Key Result

Theorem 1.1

For every $g\geq 3,$ there exist a closed Riemann surface $S$ of genus $g$ with canonical bundle $\mathcal{K}$ on which there exists abelian differentials $\phi_1,\phi_2,\phi_3,\phi_4\in H^0(S,\mathcal{K})$ such that, for real $R>0$ sufficiently large, the Higgs bundle in the $\textrm{PSL}(4,\mathbb determines a Hitchin representation $\rho:\pi_1(S)\to \textrm{PSL}(4,\mathbb R)$ together with a $\

Theorems & Definitions (93)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1: Theorem X.2.6 in KN
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 83 more