Weakly-Einstein Hermitian surfaces (Q1805934)

The authors study compact Hermitian surfaces \((M,J,h)\) whose traceless Ricci tensor is \(J\)-anti-invariant and the metric \(h\) is locally conformally Kähler. These surfaces are sometimes called \(\ast\)-Einstein and it is a consequence from the Riemannian Goldberg-Sachs theorem [\textit{M. Prazanowski} and \textit{B. Broda}, Acta Phys. Pol. B14, 637-661 (1983), \textit{V. Apostolov} and \textit{P. Gauduchon}, Int. J. Math. 8, 421-439 (1997; Zbl 0891.53054)] that every compact Einstein Hermitian surface is \(\ast\)-Einstein. The authors present large families of compact \(\ast\)-Einstein surfaces which are not Einstein. Their main result gives a description of the compact \(\ast\)-Einstein Hermitian surfaces of constant scalar and \(\ast\)-scalar curvature. The author show that any such surface \((M,J,h)\) is either Kähler-Einstein or it is a conformally flat Hopf surface and the metric \(h\) is obtained by a suitable modification of the Vaisman metric.