Harmonic maps of nonorientable surfaces to four-dimensional manifolds
In the present paper, it will be tried to construct harmonic maps from nonorientable surfaces into 4-dimensional Riemannian manifolds. Nonorientable surfaces \(M\) which are considered are quotients of Riemann surfaces \(\widehat M\) by the equivalence relation \(z\sim I(z)\), where \(I\) is an anti-holomorphic involution of \(\widehat M\) without fixed points. Especially, a projective plane and hyperelliptic Riemann surface are treated. Let \(\pi: \widehat M \to M\) be the natural projection of \(\widehat M\) to the quotient space. A map \(\widehat h\) of \(\widehat M\) into a Riemannian manifold \(N\) is factored as \(\widehat h = h \cdot \pi\), where \(h\) is a map of \(M\) into \(N\), if and only if \(\widehat hI = \widehat h\). Hence harmonic maps \(\widehat h: \widehat M \to N\) with \(\widehat h \cdot I = \widehat h\) are investigated. Let \(N\) be a 4-dimensional oriented Riemannian manifold and \(S\) its twistor space with almost complex structures \(J_ 1\) and \(J_ 2\). By introducing a natural involution \(I_ S\) of \(S\) which is anti- holomorphic with respect to \(J_ 1\) and \(J_ 2\) and using the twistor lifts \(\widetilde h: M \to S\) of Eells and Salamon for harmonic maps \(\widehat h: \widehat M \to N\), it is shown: The assignment \(\widehat h \mapsto \widetilde h\) is a bijective correspondence between the set of nonconstant conformal harmonic maps \(\widehat h: \widehat M \to N\) with \(\widehat h\cdot I = \widehat h\) and the set of nonvertical \(J_ 2\)- holomorphic curves \(\widetilde h: M\to S\) with \(\widetilde h\cdot I = I_ S \cdot \widetilde h\). Now, let \(N\) be the unit 4-sphere \(S^ 4\). Then its twistor space is the complex projective 3-space \(CP^ 3\). Bryant gives the correspondence between the set of meromorphic functions \(f\) and \(g\) on \(M\) with \(g\) nonconstant on \(\widehat M\) and the set of horizontal and holomorphic maps of \(\widehat M\) to \(CP^ 3\). It is shown: A conformal isotropic harmonic map \(\widehat h: \widehat M \to S^ 4\) has the property \(\widehat h \cdot I = \widehat h\) if and only if corresponding meromorphic functions \(f\) and \(g\) for \(\widehat h\) satisfy \(2fg^*-(gg^*+1)df/dg=0\) and \(4ff^*+(1+gg^*)^ 2=0\) where \(f^*=fI\) and \(g^*=\overline g\overline I\). Using this result, the author constructs harmonic maps \(\widehat h\) of \(S^ 2\) or hyperelliptic Riemann surfaces into \(S^ 4\) with \(\widehat h\cdot I = \widehat h\) explicitly.
scientific article; zbMATH DE number 203567
scientific article