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A Formula for the Pluricomplex Green Function of the Bidisk

Jesse J. Hulse

TL;DR

This work provides an explicit, region-based formula for the pluricomplex Green function on the bidisk with two equal-weight poles, leveraging the Lempert function equality on the bidisk. Region 1 yields a simple closed form, while Region 2 requires a hypersurface decomposition indexed by unimodular constants $e^{i\theta}$, with the Green function determined up to a constant solving a sixth-degree polynomial. The authors also translate these results into an explicit Carathéodory metric formula on the symmetrized bidisk, aligning with Agler–Young’s framework but replacing the supremum over unimodular constants by a polynomial-root condition. Region 3 and boundary cases are analyzed to complete the picture, and the findings illuminate the interplay between extremal disks, Nevanlinna–Pick interpolation, and invariant distances in related domains. Overall, the paper provides concrete, implementable expressions for Green functions and their metric consequences in two-variable complex analysis settings, with clear connections to symmetry reductions and known results on the symmetrized bidisk.

Abstract

In this paper, we derive a formula for the pluricomplex Green function of the bidisk with two poles of equal weights. In 2017, Kosiński, Thomas, and Zwonek proved the Lempert function and the pluricomplex Green function are equal on the bidisk, and their description of Lempert function was pivotal in computing the formula for the pluricomplex Green function. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. On each hypersurface, the formula is explicit up to a unimodular constant that is the root of a sixth degree polynomial. This derived formula for the bidisk leads to an explicit formula for the Carathéodory metric on the symmetrized bidisk up to a fourth degree polynomial. In 2004, Agler and Young found a formula for Carathéodory metric for the symmetrized bidisk that involves a supremum over the unimodular constants. The formula derived in this paper matches Agler and Young's formula, but the unimodular constant is determined by a 4th degree polynomial instead of the before mentioned supremum.

A Formula for the Pluricomplex Green Function of the Bidisk

TL;DR

This work provides an explicit, region-based formula for the pluricomplex Green function on the bidisk with two equal-weight poles, leveraging the Lempert function equality on the bidisk. Region 1 yields a simple closed form, while Region 2 requires a hypersurface decomposition indexed by unimodular constants , with the Green function determined up to a constant solving a sixth-degree polynomial. The authors also translate these results into an explicit Carathéodory metric formula on the symmetrized bidisk, aligning with Agler–Young’s framework but replacing the supremum over unimodular constants by a polynomial-root condition. Region 3 and boundary cases are analyzed to complete the picture, and the findings illuminate the interplay between extremal disks, Nevanlinna–Pick interpolation, and invariant distances in related domains. Overall, the paper provides concrete, implementable expressions for Green functions and their metric consequences in two-variable complex analysis settings, with clear connections to symmetry reductions and known results on the symmetrized bidisk.

Abstract

In this paper, we derive a formula for the pluricomplex Green function of the bidisk with two poles of equal weights. In 2017, Kosiński, Thomas, and Zwonek proved the Lempert function and the pluricomplex Green function are equal on the bidisk, and their description of Lempert function was pivotal in computing the formula for the pluricomplex Green function. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. On each hypersurface, the formula is explicit up to a unimodular constant that is the root of a sixth degree polynomial. This derived formula for the bidisk leads to an explicit formula for the Carathéodory metric on the symmetrized bidisk up to a fourth degree polynomial. In 2004, Agler and Young found a formula for Carathéodory metric for the symmetrized bidisk that involves a supremum over the unimodular constants. The formula derived in this paper matches Agler and Young's formula, but the unimodular constant is determined by a 4th degree polynomial instead of the before mentioned supremum.

Paper Structure

This paper contains 26 sections, 18 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Region One
  4. Region 2: the Case When $z_1=0$
  5. General Computations
  6. An Explicit formula when $z$ is real valued.
  7. The Case When $v=i$ and $\omega =-1$
  8. The Case When $v=1$ and $\omega=-1$
  9. An Explicit Formula When $e^{i\theta}=\pm i$
  10. The extremal disks for $e^{i\theta}=i$
  11. The extremal disks for $e^{i\theta}=-i$
  12. A Family of Hypersurfaces Spanning Region 2
  13. A Few Preliminary Results.
  14. An Equation for $g_{\mathop{\mathrm{\mathbb D}}\limits^2}$ Along an Extremal Disk
  15. Eliminating the parameter $z$
  16. ...and 11 more sections

Key Result

Proposition 4.1

Write $c=rv$ where $r>0$ and $|v|=1$. Conditions (blaschkecon1)-(blaschkecon5) imply the following: where

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Lemma 4.4
  • ...and 29 more