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A note On the existence of solutions to Hitchin's self-duality equations

Yu Feng, Shuo Wang, Bin Xu

Abstract

In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these equations starting from a Higgs stable bundle. In this paper, we fill in some technical details in Hitchin's original proof by the following three steps. First, we reduce the existence of a solution of class $L_1^2$ to minimizing the energy functional within a Higgs stable orbit of the $L_2^2$ complex gauge group action. Second, using this transformation, we obtain a solution of class $L_1^2$ in this orbit. These two steps primarily follow Hitchin's original approach. Finally, using the Coulomb gauge, we construct a smooth solution by applying an $L_2^2$ unitary gauge transformation to the $L_1^2$ solution constructed previously. This last step provides additional technical details to Hitchin's original proof.

A note On the existence of solutions to Hitchin's self-duality equations

Abstract

In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these equations starting from a Higgs stable bundle. In this paper, we fill in some technical details in Hitchin's original proof by the following three steps. First, we reduce the existence of a solution of class to minimizing the energy functional within a Higgs stable orbit of the complex gauge group action. Second, using this transformation, we obtain a solution of class in this orbit. These two steps primarily follow Hitchin's original approach. Finally, using the Coulomb gauge, we construct a smooth solution by applying an unitary gauge transformation to the solution constructed previously. This last step provides additional technical details to Hitchin's original proof.
Paper Structure (25 sections, 26 theorems, 93 equations)

This paper contains 25 sections, 26 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Background and Main Result
  3. Self-duality equations
  4. Self-duality equations on principal bundles
  5. Self-duality equations on associated bundles
  6. Sobolev Case
  7. Gauge Groups
  8. Smooth Case
  9. Sobolev case
  10. Higgs stable pair
  11. Moment Map
  12. Atiyah-Bott's Result
  13. Main Result
  14. Reduction of Solving Self-Duality Equations
  15. Proof of Proposition \ref{['eq-conditions']}
  16. ...and 10 more sections

Key Result

Lemma 2.1

Denote by $L_k^p$ the space of $L_k^p$ functions defined on the unit disk in $\mathbb{C}$. Then Let $E$ be a vector bundle over a compact Riemann surface $M$. Denote by $L_k^p(M, E)$ the vector space of $L_k^p$ sections of $E$. Then

Theorems & Definitions (50)

  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 40 more