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This is the first part in a series of three articles in which are studied thedomains of monogenicity for the $n$-Cauchy-Fueter operator. Using the twistortheory, we will in this article show that for a given open subset $U$ of$\mathbb{Q}^n$, there is an open subset $\mathcal{H}(U)$, called the monogenichull of $U$, of $M_{2n\times 2}^{\mathbb{C}}=\mathbb{Q}^n\otimes\mathbb{C}$such that each monogenic function in $U$ extends to a unique pair ofholomorphic functions on $\mathcal{H}(U)$. In the second part of the series wewill exploit the twistor theory furthermore to prove that any pseudoconvexdomain in $\mathbb{Q}^n$ is a domain of monogenicity. In the third part of theseries, we show the other implication and provide a geometric characterizationof the domains of monogenicity.

Monogenic hull for the n-Cauchy-Fueter operator and twistor theory