Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Domination between non-Fuchsian representations and anti-de Sitter geometry

Farid Diaf, Abderrahim Mesbah, Nathaniel Sagman

TL;DR

The paper develops a framework linking domination between non-Fuchsian surface group representations to geometric structures on 3-manifolds via harmonic maps. It introduces a holomorphic data parameterization $(\phi,D)$ for harmonic maps, establishes sharp energy-inequality criteria for domination, and proves that domination by branched immersions yields a family of dominating representations whose Euler numbers satisfy $\mathrm{eu}(j)=\mathrm{eu}(\rho)+k$. These domination data naturally construct singular anti-de Sitter 3-manifolds (spin-cone AdS) and, in the branched-immersion case, branched AdS manifolds, by a gluing procedure that encodes the singular locus as the image of the branching set. The results extend the Deroin–Tholozan program to non-Fuchsian targets, providing a robust bridge from harmonic map theory and Higgs-bundle data to concrete AdS geometric structures and their holonomy representations, with wide potential for further generalizations to length-spectrum questions, higher rank, and richer singular AdS geometries.

Abstract

Motivated by work of various authors on domination between surface group representations, harmonic maps, and $3$-dimensional anti-de Sitter geometry, we study a new domination problem between non-Fuchsian representations of closed surface groups. We solve the problem for representations that admit branched harmonic immersions, and we show that, outside of this case, the problem cannot always be solved. We then show that a dominating pair gives rise to an anti-de Sitter $3$-manifold with singularities, and we construct large families of branched anti-de Sitter $3$-manifolds.

Domination between non-Fuchsian representations and anti-de Sitter geometry

TL;DR

The paper develops a framework linking domination between non-Fuchsian surface group representations to geometric structures on 3-manifolds via harmonic maps. It introduces a holomorphic data parameterization for harmonic maps, establishes sharp energy-inequality criteria for domination, and proves that domination by branched immersions yields a family of dominating representations whose Euler numbers satisfy . These domination data naturally construct singular anti-de Sitter 3-manifolds (spin-cone AdS) and, in the branched-immersion case, branched AdS manifolds, by a gluing procedure that encodes the singular locus as the image of the branching set. The results extend the Deroin–Tholozan program to non-Fuchsian targets, providing a robust bridge from harmonic map theory and Higgs-bundle data to concrete AdS geometric structures and their holonomy representations, with wide potential for further generalizations to length-spectrum questions, higher rank, and richer singular AdS geometries.

Abstract

Motivated by work of various authors on domination between surface group representations, harmonic maps, and -dimensional anti-de Sitter geometry, we study a new domination problem between non-Fuchsian representations of closed surface groups. We solve the problem for representations that admit branched harmonic immersions, and we show that, outside of this case, the problem cannot always be solved. We then show that a dominating pair gives rise to an anti-de Sitter -manifold with singularities, and we construct large families of branched anti-de Sitter -manifolds.

Paper Structure

This paper contains 35 sections, 41 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Harmonic maps and domination
  4. Future directions
  5. Length spectrum domination
  6. Higher rank
  7. More on singular $\textrm{AdS}$ $3$-manifolds
  8. Outline of the paper
  9. Acknowledgements
  10. Preliminaries on harmonic maps
  11. Equivariant harmonic maps
  12. Energies and Hopf differentials
  13. Divisors and Higgs bundles
  14. $G$-Higgs bundles
  15. Proposition \ref{['prop: holomorphic data']}
  16. ...and 20 more sections

Key Result

Theorem A

Let $X$ be a closed Riemann surface of genus $\geq 2$ with fundamental group $\Gamma$ and let $\rho:\Gamma\to \mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ be a representation with $0< \textrm{eu}(\rho)<2g-2$ such that the unique equivariant harmonic map $f:\widetilde{X}\to (\mathbb{H}^2,\sigma)$ is

Figures (1)

  • Figure 1: The behavior of the developing map restricted to the slice $(0, 1) \times \mathbb{R} \times \{y\}$.

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Corollary 2
  • Remark 1.2
  • Theorem 1.3: Deroin-Tholozan Deroin_thlozan_domination and Tholozan Tholozan combined
  • Proposition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Theorem C
  • ...and 82 more