Harmonic almost-complex structures on twistor spaces
The main purpose of this paper is to study the harmonicity of the Atiyah-Hitchin-Singer [\textit{M. F. Atiyah, N. J. Hitchin} and \textit{I. M. Singer}, Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)] and Eells-Salamon [\textit{J. Eells} and \textit{S. Salamon}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 12, 589-640 (1985; Zbl 0627.58019)] almost-complex structures \(J_1\) and \(J_2\) on the negative twistor space \((Z_t,h_t)\) of an oriented Riemannian four-manifold, where \(h_t, t>0\) is the metric on \(Z\) obtained by rescaling \(\tilde{h}\) on the fibres by \(t\). Our main result is the following theorem, which may be considered as a variational interpretation of the self-duality condition: Theorem 1: Let \((M,g)\) be an oriented Riemannian four-manifold and \((Z_t, h_t)\) be its negative twistor space. Then, (i) The Atiyah-Hitchin-Singer almost-complex structure \(J_1\) on \((Z_t,h_t)\) is harmonic, if and only if \((M,g)\) is a self-dual manifold. (ii) The Eells-Salamon almost-complex structure \(J_2\) on \((Z_t,h_t)\) is harmonic, if and only if \((M,g)\) is a self-dual manifold with constant scalar curvature. To prove Theorem 1, we show at first, by means of the Weitzenböck formula, that the Euler-Lagrange equation is equivalent to the condition that the Laplacian of the Kähler form and its image under the curvature operator have equal \([(2,0)+(0,2)]\)-parts. In the case of the twistor space \((Z,h_t,J_n), n=1,2,\) we obtain explicit formulas for these 2-forms in terms of the curvature of the base manifold \((M,g)\) and prove that the above condition is fulfilled for the horizontal vectors of \(Z\), if and only if the base manifold is self-dual. Finally we show that if \((M,g)\) is self-dual, the Euler-Lagrange equation for \(J_1\) is automatically satisfied, whereas, for \(J_2\), it is equivalent to \((M,g)\) being of constant scalar curvature.
scientific article; zbMATH DE number 1850616
scientific article