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Epstein surfaces for $G-$opers

Joaquín Lema

TL;DR

The paper develops a higher-rank generalization of Epstein surfaces for $G$-opers, constructing Epstein surfaces Ep$(z)$ inside the symmetric space $bX$ via osculation with a fixed Fuchsian oper. It provides a concrete, normalized parametrization of the oper moduli via the Hitchin base, and establishes quantitative regularity and curvature bounds for Ep, enabling a criteria for $oldsymbol{ riangle}$-Anosov holonomy through Davalo’s nearly geodesic framework. The results yield explicit neighborhoods of Anosov opers around the Fuchsian locus, applicable to general $G$ and sharpened in cyclic cases, with consequences for domains of discontinuity and transversality. The work links developing maps, osculation maps, and the asymptotic geometry of holonomies, contributing to the understanding of Anosov representations in higher rank and their geometric structures.”

Abstract

Given a complex semisimple Lie group $G$, we introduce the notion of an Epstein surface associated to a $G$-oper. These surfaces generalize Epstein's classical construction for $G=PGL_2 (\mathbb{C})$. As an application, we provide a criterion that ensures that the holonomy of the oper is $Δ-$Anosov. Finally, we discuss how the developing map of the oper interacts with domains of discontinuity of the holonomy (whenever Anosov) and the transversality properties it satisfies. Along the way, we provide a quick review of opers that we hope serves as a self-contained introduction.

Epstein surfaces for $G-$opers

TL;DR

The paper develops a higher-rank generalization of Epstein surfaces for -opers, constructing Epstein surfaces Ep inside the symmetric space via osculation with a fixed Fuchsian oper. It provides a concrete, normalized parametrization of the oper moduli via the Hitchin base, and establishes quantitative regularity and curvature bounds for Ep, enabling a criteria for -Anosov holonomy through Davalo’s nearly geodesic framework. The results yield explicit neighborhoods of Anosov opers around the Fuchsian locus, applicable to general and sharpened in cyclic cases, with consequences for domains of discontinuity and transversality. The work links developing maps, osculation maps, and the asymptotic geometry of holonomies, contributing to the understanding of Anosov representations in higher rank and their geometric structures.”

Abstract

Given a complex semisimple Lie group , we introduce the notion of an Epstein surface associated to a -oper. These surfaces generalize Epstein's classical construction for . As an application, we provide a criterion that ensures that the holonomy of the oper is Anosov. Finally, we discuss how the developing map of the oper interacts with domains of discontinuity of the holonomy (whenever Anosov) and the transversality properties it satisfies. Along the way, we provide a quick review of opers that we hope serves as a self-contained introduction.
Paper Structure (37 sections, 49 theorems, 107 equations, 4 figures)

This paper contains 37 sections, 49 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Ahlfors-Weill Theorem
  3. Generalized Epstein surfaces
  4. Main results and Open Questions
  5. Related work
  6. Structure of the article
  7. Notation and Conventions
  8. Lie Theory Preliminaries
  9. Full flag manifold
  10. Symmetric spaces
  11. Principal triplets
  12. Opers: definition and parametrization
  13. Developing data
  14. Definitions and basic properties
  15. The Drinfeld-Sokolov normal form
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $X$ be a compact hyperbolic Riemann surface, and identify $\mathcal{P}(X)$ with $H^0 (X,K^2)$ using the Fuchsian projective structure as a basepoint. Then, the complex projective structure associated to an $\alpha \in H^0 (X,K^2)$ satisfying $\lVert\alpha\rVert < \frac{1}{2}$ will have quasi-Fuc

Figures (4)

  • Figure 1: Directions tangent to $T^1 \mathcal{F}_n$ slide $F_i$ into $F_{i+1}$
  • Figure 2: When $G = SL_3 (\mathbb{C})$, the boundary stratifies into $G$ orbits identified with the full flag manifold $\mathcal{F}_3$ or partial flag manifolds like $\mathbb{P}^2$
  • Figure 3: Decomposition induced on $\mathfrak{sp}_6$ by a principal triple (exponents $1,3$ and $5$). Each dot corresponds to a weight space for $h$.
  • Figure 4: The osculating map allows us to propagate the data at infinity down to the symmetric space.

Theorems & Definitions (106)

  • Theorem 1.1: ahlfors1962uniqueness
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.8
  • Definition 1.9
  • Definition 1.11
  • Theorem 1.12
  • ...and 96 more