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

TL;DR

This paper establishes the transformation properties of the $k$-Cauchy-Fueter complex under quaternionic projective transformations, enabling new geometric and analytical applications in quaternionic analysis and pluripotential theory.

Contribution

It explicitly derives the transformation formulae of the $k$-Cauchy-Fueter complex under ${ m SL}(n+1, ext{H})$ and constructs invariant operators, advancing quaternionic analysis and geometry.

Findings

Transformation formulae for $k$-Cauchy-Fueter complex under quaternionic fractional linear transformations.

Construction of the complex over locally projective flat manifolds.

Introduction of a quaternionic projectively invariant operator from the Monge-Ampère operator.

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-Amp\`{e}re operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.