Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2210.13902 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917585219485696 |
|---|---|
| author | Wang, Wei |
| author_facet | Wang, Wei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13902 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quaternionic projective invariance of the $k$-Cauchy-Fueter complex and applications I Wang, Wei Complex Variables Differential Geometry 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. |
| title | Quaternionic projective invariance of the $k$-Cauchy-Fueter complex and applications I |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2210.13902 |