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Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions

Sebastian Heller, Lothar Schiemanowski, Hartmut Weiss

TL;DR

The paper develops a geometric duality between harmonic maps into $\mathbb{H}^3$ and $\operatorname{dS}_3$, and a corresponding gauge-theoretic link between $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ self-duality equations. It introduces transgressive harmonic maps into $\mathbb{S}^3$, whose oblique Gauss maps extend to $\operatorname{dS}_3$ with rank drop along a crossing set, and it embeds these ideas into the Deligne–Hitchin moduli framework via real holomorphic sections. The authors develop a gluing construction that produces large families of singular and regular solutions by patching model, fiducial, and limiting configurations, analyzed through 0-calculus and semiclassical techniques. As an application, they construct $\tau$-real negative sections of arbitrarily large energy in the Deligne–Hitchin moduli space that are not twistor lines, thereby connecting non-compact Higgs data to singular harmonic maps and extending non-abelian Hodge theory to singular settings with controlled asymptotics.

Abstract

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\operatorname{dS}_3$, best viewed as a generalized Gauss map. On the gauge theoretic side, it matches SU(2) and SU(1,1) solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of transgressive harmonic maps $f:M\to \mathbb{S}^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $τ$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions

TL;DR

The paper develops a geometric duality between harmonic maps into and , and a corresponding gauge-theoretic link between and self-duality equations. It introduces transgressive harmonic maps into , whose oblique Gauss maps extend to with rank drop along a crossing set, and it embeds these ideas into the Deligne–Hitchin moduli framework via real holomorphic sections. The authors develop a gluing construction that produces large families of singular and regular solutions by patching model, fiducial, and limiting configurations, analyzed through 0-calculus and semiclassical techniques. As an application, they construct -real negative sections of arbitrarily large energy in the Deligne–Hitchin moduli space that are not twistor lines, thereby connecting non-compact Higgs data to singular harmonic maps and extending non-abelian Hodge theory to singular settings with controlled asymptotics.

Abstract

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space and harmonic maps from Riemann surfaces to de Sitter three-space , best viewed as a generalized Gauss map. On the gauge theoretic side, it matches SU(2) and SU(1,1) solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of transgressive harmonic maps : these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain -real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Paper Structure

This paper contains 39 sections, 72 theorems, 466 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic geometry and the conformal geometry of $\mathbb S^3$
  4. Gauge-theoretic background
  5. Harmonic maps from surfaces to $\operatorname{dS}_3$
  6. The matrix model of $\operatorname{dS}_3$
  7. Equivariant harmonic maps into $\operatorname{dS}_3$
  8. $\lambda$-connections and the Deligne--Hitchin moduli space
  9. Hitchin self-duality equations and transgressive harmonic maps
  10. Oblique Gauß maps for harmonic maps into $\mathbb H^3$
  11. The dual map
  12. The conformally invariant Hopf differential
  13. Transgressive harmonic maps
  14. The twist construction
  15. The model solution
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $M$ be a compact Riemann surface of genus at least $2$ and let $(E, \overline{\partial}_E, \varphi)$ be a Higgs bundle with $\deg E = 0$. Suppose that $q = \det \varphi$ has simple zeros and contains at least one Strebel cylinder (see Section sec:41). Let $M^\vee$ denote the complement of the co These solutions induce harmonic, equivariant maps $f_{t,\operatorname{hyp}}$ into $\mathbb H^3_\pm$

Theorems & Definitions (145)

  • Theorem 1.1: Theorem \ref{['thm:existence_hoelder_transgressive_maps']}, Corollary \ref{['cor:hoelder_dual_maps']}
  • Theorem 1.2: Theorem \ref{['thm:existence_smooth_transgressive_maps']}
  • Theorem 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 135 more