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Equivariant finite energy proper minimal surfaces in $\mathbb{CH}^2$

Indranil Biswas, Pradip Kumar, John Loftin

Abstract

Given a noncompact Riemann surface $Σ_0\,=\, Σ\setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $Σ$, and a reductive representation $ρ\,:\,π_1(Σ_0)\,\longrightarrow\, \mathrm{PU}(2,1)$, we prove that any finite--energy $ρ$--equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of $ρ$ is parabolic. Assuming parabolic peripheral holonomy, we give an explicit parametrization of complete finite--energy immersions in the mixed case in terms of tame parabolic $\mathrm{PU}(2,1)$--Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of $ρ$ equivariant proper $\mathbb{CH}^2$ $n$--noids on $\mathbb{CP}^1\setminus P$ for $|P|\,\ge\, 5$.

Equivariant finite energy proper minimal surfaces in $\mathbb{CH}^2$

Abstract

Given a noncompact Riemann surface , where is a finite subset of a compact connected Riemann surface , and a reductive representation , we prove that any finite--energy --equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of is parabolic. Assuming parabolic peripheral holonomy, we give an explicit parametrization of complete finite--energy immersions in the mixed case in terms of tame parabolic --Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of equivariant proper --noids on for .
Paper Structure (35 sections, 18 theorems, 131 equations)

This paper contains 35 sections, 18 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Complex hyperbolic Space
  3. The projective model and Bergman metric
  4. Totally geodesic submanifolds and rank
  5. Isometries of $\mathbb{CH}^2$
  6. Classification of elements of $\mathrm{PU}(2,1)$
  7. Curvature and CAT$(0)$ features
  8. Proper minimal immersion with finite energy
  9. Type decomposition of $df$
  10. Isometry and properness
  11. Around a cusp
  12. Properness enforces parabolic peripheral holonomy
  13. Parabolic $\mathrm{PU}(2,1)$ Higgs bundles
  14. Parabolic $\mathrm{U}(2,1)$ Higgs data
  15. Peripheral holonomy, weights, and nilpotent residues
  16. ...and 20 more sections

Key Result

Proposition 4.1

Let $M\,=\,\widetilde{M}/\Gamma$ be a complete, finite--volume hyperbolic surface, and let $(X,\,g)$ be a $\mathrm{CAT}(-1)$ Hadamard manifold. For a representation $\rho_1\,:\,\Gamma\,\longrightarrow\, \mathrm{Iso}(X,g)$, there exists a finite--energy $\rho_1$--equivariant map $u\,:\,\widetilde{M}\

Theorems & Definitions (40)

  • Remark 2.1
  • Definition 3.2: Properness around a puncture
  • Definition 3.3: Completeness around a puncture
  • Proposition 4.1: Sagman2023
  • Corollary 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 30 more