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Harmonic forms on ALE Ricci-flat 4-manifolds

Gao Chen, Hao Yan

Abstract

In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.

Harmonic forms on ALE Ricci-flat 4-manifolds

Abstract

In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.

Paper Structure

This paper contains 30 sections, 3 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The settings of ALE-manifolds
  4. Harmonic functions
  5. Case 1
  6. Case 2
  7. Determine the constant
  8. Case 1
  9. Case 2
  10. Harmonic 1-forms
  11. Find the expansions
  12. Determine the relation of the constants
  13. The Constants
  14. Type I
  15. Type II
  16. ...and 15 more sections

Key Result

Theorem 1.1

We fix $\epsilon\in(0,1)$. For any $a_{ij}\in \mathbb{R}$, $i,j=1,\cdots,4$, such that $a_{ij}=a_{ji}$ and $\sum_{i,j=1}^{4} a_{ij}x^i x^j$ is invariant under $\Gamma$, there exists a unique smooth function $u_a$ on $X$ such that for all $k\ge 0$ (see Def defnorm), and $\Delta_X u_a =(d\delta_X+\delta_X d)u_a= -2\sum_{i,j=1}^{4} a_{ij}\delta_{ij}$ on $X$. Moreover, the expansion of $u_a$ is given

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3