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The extended Bogomolny equations on $\mathbb {R}^2 \times \mathbb {R}^+$ I: Nilpotent Higgs field

Panagiotis Dimakis

Abstract

We study the extended Bogomolny equations with gauge group $SU(2)$ on $\mathbb {R}^2 \times \mathbb {R}^+$ with generalized Nahm pole boundary conditions and nilpotent Higgs field. We completely classify solutions by relating them to certain holomorphic data through a Kobayashi-Hitchin correspondence.

The extended Bogomolny equations on $\mathbb {R}^2 \times \mathbb {R}^+$ I: Nilpotent Higgs field

Abstract

We study the extended Bogomolny equations with gauge group on with generalized Nahm pole boundary conditions and nilpotent Higgs field. We completely classify solutions by relating them to certain holomorphic data through a Kobayashi-Hitchin correspondence.
Paper Structure (16 sections, 12 theorems, 96 equations)

This paper contains 16 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Setting up the problem
  4. The Extended Bogomolny Equations
  5. Boundary and asymptotic conditions
  6. The holomorphic line sub-bundle
  7. The approximate solution
  8. Linear Analysis
  9. The linearized operator
  10. Mapping properties of $\mathcal{L}_H$
  11. Improving the approximate solution
  12. The continuity method
  13. Set up and openness
  14. A priori estimates
  15. Existence
  16. ...and 1 more sections

Key Result

Theorem 1.3

There is a bijective correspondence between $\operatorname{EBE}(N,K)$ and $\{(\mathcal{E}, P(z), \mathcal{L}, K)\}$.

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 3.1
  • Theorem 3.2
  • ...and 18 more