Twistor transforms of quaternionic functions and orthogonal complex structures. (English) Zbl 1310.53045

The authors study orthogonal complex structures (OCS) on the space \((\mathbb R^4=\mathbb H,\mathrm{can})\) with the standard flat metric can. Let \(S=\{q\in\mathbb H:q^2=-1\}\) be the sphere of imaginary unit quaternions. An orthogonal almost complex structure defined on an open subset \(\Omega\subset\mathbb R^4=\mathbb H\) is simply a map \(J:\Omega\to S\). The authors define a special OCS structure \(\mathbb J\) on \(\Omega=\mathbb H-\mathbb R\) as \(\mathbb J_q=\frac{\mathrm{Im}(q)}{|\mathrm{Im}(q)|}\in S\). Let \(\Omega\) be a domain in \(\mathbb H\) and for \(I\in S\) let us denote \(\Omega_I=\Omega\cap L_I\) where \(L_I=\mathbb R+I\mathbb R\). A function of the quaternion variable \(f:\Omega\to\mathbb H\) is called regular if for all \(I\in S\) the restriction \(f_{\Omega_I}\) is holomorphic. Let \(\Omega\) be a symmetric slice domain and let \(f:\Omega\to\mathbb H\) be an injective regular function. Then \(f\) induces an OCS structure \(\mathbb J^f=f_*\mathbb Jf_*^{-1}\) on \(f(\Omega-\mathbb R)\). The authors study OCS induced by regular functions \(f\) of the quaternion variable. The twistor transform of \(f\) corresponds to a holomorphic curve in a Klein quadric. “The case in which \(\Omega\) is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space \(\mathbb {CP}^3\).”


53C28 Twistor methods in differential geometry
30G35 Functions of hypercomplex variables and generalized variables
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J26 Rational and ruled surfaces
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