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Quaternionic projective invariance of the $k$-Cauchy-Fueter complex and applications I

Wei Wang

Abstract

The $k$-Cauchy-Fueter complex in quaternionic analysis is the counterpart of the Dolbeault complex in complex analysis. In this paper, we find the explicit transformation formula of these complexes under ${\rm SL}(n+1,\mathbb{H})$, which acts on $\mathbb{H}^{ n}$ as quaternionic fractional linear transformations. These transformation formulae have several interesting applications to $k$-regular functions, the quaternionic counterpart of holomorphic functions, and geometry of domains. They allow us to construct the $k$-Cauchy-Fueter complex over locally projective flat manifolds explicitly and introduce various notions of pluripotential theory on this kind of manifolds. We also introduce a quaternionic projectively invariant operator from the quaternionic Monge-Ampère operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.

Quaternionic projective invariance of the $k$-Cauchy-Fueter complex and applications I

Abstract

The -Cauchy-Fueter complex in quaternionic analysis is the counterpart of the Dolbeault complex in complex analysis. In this paper, we find the explicit transformation formula of these complexes under , which acts on as quaternionic fractional linear transformations. These transformation formulae have several interesting applications to -regular functions, the quaternionic counterpart of holomorphic functions, and geometry of domains. They allow us to construct the -Cauchy-Fueter complex over locally projective flat manifolds explicitly and introduce various notions of pluripotential theory on this kind of manifolds. We also introduce a quaternionic projectively invariant operator from the quaternionic Monge-Ampère operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.
Paper Structure (15 sections, 19 theorems, 208 equations)

This paper contains 15 sections, 19 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. The complexified version of the $k$-Cauchy-Fueter complex
  3. ${\rm SL}( n+1, \mathbb{H})$ and its complexification
  4. ${\rm G}/{\rm P}$
  5. The ${\rm SL}(2n+2, \mathbb{C})$-invariance of the complexified version
  6. Proof of the invariance for the case $j<k$
  7. Proof of the invariance for the case $j> k$
  8. Proof of the invariance for the case $k=j$
  9. The invariance on $\mathbb{H}^{ n }$ and complexes on locally projective flat manifolds
  10. The invariance on $\mathbb{H}^{ n }$
  11. Complexes over locally projective flat manifolds
  12. The quaternionic Monge-Ampère operator on locally projective flat manifolds
  13. The cone bundle $\text{SP}^{2p}E^*$ of strongly positive $2p$-elements
  14. Closed positive currents and "integrals"
  15. Projective invariance

Key Result

Theorem 1.1

$\mathcal{D}_j$ is ${\rm SL}(n+1,\mathbb{H})$-invariant, i.e. for any $g\in {\rm SL}(n+1,\mathbb{H})$ and $f\in \Gamma(\mathbb{H}^{ n}, \mathcal{V}_j )$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.2
  • Remark 2.3
  • ...and 29 more