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Twistor construction of some multivalued harmonic functions on ${\bf R}^{3}$

Simon Donaldson

TL;DR

This work builds explicit multivalued harmonic functions on ${\mathbb R}^{3}$ with an ellipse as the branch set by a twistor-integral construction. A one-parameter family, indexed by $\epsilon\in(-1,1)$, is obtained via a cohomological data $F$ on the twistor space, with $F$ involving $Q(w)=1+\tfrac{\epsilon}{2}(w^{2}+w^{-2})$ and $\tan^{-1}$, yielding wall-matching and $Z$-odd symmetry. The resulting solutions have quadratic asymptotics at infinity with coefficients $\lambda(\epsilon),\mu(\epsilon),\nu(\epsilon)$ satisfying $\lambda+\mu=\nu$ and are connected to a genus-1 curve $\Sigma_{\epsilon}$ in the twistor space; the approach highlights a deep link between elliptic geometry, twistor theory, and explicit harmonic analysis, suggesting pathways to new PDE constructions in low dimensions. The analysis also clarifies the complex-geometry picture and provides asymptotic and monotonicity properties of the defining coefficients, with special attention to the rotationally symmetric case $\epsilon=0$.

Abstract

In this paper twistor methods are used to construct a family of multivalued harmonic functions on ${\bf R}^{3}$ which were obtained by Dashen Yan using different methods. The branching sets for the solutions are ellipses and the functions have quadratic growth at infinity.

Twistor construction of some multivalued harmonic functions on ${\bf R}^{3}$

TL;DR

This work builds explicit multivalued harmonic functions on with an ellipse as the branch set by a twistor-integral construction. A one-parameter family, indexed by , is obtained via a cohomological data on the twistor space, with involving and , yielding wall-matching and -odd symmetry. The resulting solutions have quadratic asymptotics at infinity with coefficients satisfying and are connected to a genus-1 curve in the twistor space; the approach highlights a deep link between elliptic geometry, twistor theory, and explicit harmonic analysis, suggesting pathways to new PDE constructions in low dimensions. The analysis also clarifies the complex-geometry picture and provides asymptotic and monotonicity properties of the defining coefficients, with special attention to the rotationally symmetric case .

Abstract

In this paper twistor methods are used to construct a family of multivalued harmonic functions on which were obtained by Dashen Yan using different methods. The branching sets for the solutions are ellipses and the functions have quadratic growth at infinity.

Paper Structure

This paper contains 7 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Proof of Theorem 1
  4. Further discussion and results
  5. Existence and uniqueness
  6. Complex geometry
  7. Evaluating some integrals

Key Result

Theorem 1

For any $\epsilon$ with $-1<\epsilon<1$ set $a=\sqrt{1+\epsilon}, b=\sqrt{1-\epsilon}$. There is a harmonic function $\phi$ on ${\bf R}^{3}\setminus \overline{W_{\infty}}$ satisfying the matching conditions (1) and such that

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Lemma 4
  • Proposition 2