论文标题
$ K $ -CAUCHY-FUETER COMPLEX和应用I
Quaternionic projective invariance of the $k$-Cauchy-Fueter complex and applications I
论文作者
论文摘要
$ K $ -Cauchy-fueter在Quaternionic Analiveration中是复杂分析中Dolbeault Complex的对应物。在本文中，我们找到了这些复合物在$ {\ rm sl}下的明确转换公式（n+1，\ mathbb {h}）$，该$在$ \ mathbb {h}^{n} $上作为quaternionic fractional fractional线性变换。这些转换公式在$ k $ regormard功能，全体形态函数的Quaternionic对应物和域的几何形状上具有多个有趣的应用。它们使我们能够明确地构建$ k $ cauchy-fueter综合体，并明确地构建了本地投射的平面歧管，并在这种歧管上介绍了多能理论的各种概念。我们还从Quaternionic Monge-Ampère操作员引入了一个Quaternionic的投影不变的操作员，该操作员可用于在复杂的分析中概括Fefferman的构造，以概述域名的域名定义密度。
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.
