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Isolated singularities of Toda equations and cyclic Higgs bundles

Qiongling Li, Takuro Mochizuki

Abstract

This paper is the second part of our study on the Toda equations and the cyclic Higgs bundles associated to $r$-differentials over non-compact Riemann surfaces. We classify all the solutions up to boundedness around the isolated singularity of an $r$-differential under the assumption that the $r$-differential is meromorphic or has some type of essential singularity. As a result, for example, we classify all the solutions on ${\mathbb C}$ if the $r$-differential is a finite sum of the exponential of polynomials.

Isolated singularities of Toda equations and cyclic Higgs bundles

Abstract

This paper is the second part of our study on the Toda equations and the cyclic Higgs bundles associated to -differentials over non-compact Riemann surfaces. We classify all the solutions up to boundedness around the isolated singularity of an -differential under the assumption that the -differential is meromorphic or has some type of essential singularity. As a result, for example, we classify all the solutions on if the -differential is a finite sum of the exponential of polynomials.

Paper Structure

This paper contains 129 sections, 141 theorems, 390 equations.

Table of Contents

  1. Introduction
  2. Harmonic bundles and Toda equations
  3. Higgs bundles associated with $r$-differentials and harmonic metrics
  4. Toda equations associated with $r$-differentials
  5. Existence and uniqueness of complete solutions
  6. Uniqueness and non-uniqueness of general solutions
  7. Remark on the difference of the conventions for the induced Hermitian metrics
  8. Isolated singularities
  9. The case of poles
  10. Holomorphic functions with multiple growth orders
  11. The case where $q$ is not meromorphic but has multiple growth orders
  12. Global classification in the case where $q$ has multiple growth orders
  13. Comparison of the methods and the results
  14. A generalization
  15. Examples on ${\mathbb C}$
  16. ...and 114 more sections

Key Result

Theorem 1.4

Let $X$ be a non-compact Riemann surface with a holomorphic $r$-differential $q$. Assume either (i) $X$ is hyperbolic, or (ii) $X$ is parabolic and $q\neq 0$. Here, we say that $X$ is hyperbolic (resp. parabolic) if a universal covering of $X$ is the upper half plane (resp. ${\mathbb C}$). Then, the

Theorems & Definitions (178)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Note0
  • Proposition 1.5: Note0
  • Remark 1.6
  • Proposition 1.7: Note0
  • Remark 1.8
  • Theorem 1.9: Theorem \ref{['thm;20.6.9.30']}
  • Theorem 1.10
  • ...and 168 more