On the boundary complex of the $k$-Cauchy-Fueter complex

TL;DR

This paper constructs boundary complexes for the $k$-Cauchy-Fueter complex in quaternionic analysis, enabling new extension theorems and introducing a quaternionic Monge-Ampère operator on specific hypersurfaces.

Contribution

It explicitly constructs boundary complexes for $k$-Cauchy-Fueter complexes and applies them to establish extension theorems and define a quaternionic Monge-Ampère operator.

Findings

Established Hartogs-Bochner extension for $k$-regular functions.

Constructed boundary complexes on quadratic hypersurfaces with nilpotent Lie group structure.

Introduced quaternionic Monge-Ampère operator and generalized vanishing theorems.

Abstract

The $k$-Cauchy-Fueter complex, $k=0,1,\ldots$, in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis.They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for $k$-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge-Amp\`{e}re operator and opens the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange's vanishing theorem and Hartogs-Bochner extension for $k$-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.