HOLOMORPHIC FUNCTIONS AND REGULAR QUATERNIONIC FUNCTIONS ON THE HYPERKÄHLER SPACE ℍ

Abstract

Let H be the space of quaternions, with its standard hypercomplex structure. Let $\mathcal R(\Omega)$ be the module of \emph{$\psi$-regular} functions on $\Omega$. For every $p\in H$, $p^2=-1$, $\mathcal R(\Omega)$ contains the space of holomorphic functions w.r.t. the complex structure $J_p$ induced by $p$. We prove the existence, on any bounded domain $\Omega$, of $\psi$-regular functions that are not $J_p$-holomorphic for any $p$. Our starting point is a result of Chen and Li concerning maps between hyperk\ahler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes $J_p$-holomorphic functions among $\psi$-regular functions.