2022/10/25 by Wei Wang, Wang, Wei · 1 citation
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2210.13902
openalex publication_date 2022/10/25 · openalex created_date 2022/11/01 · openalex updated_date 2026/07/28
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,ℍ), which acts on 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.