 
Summary: Ann. Inst. Fourier, Grenoble
49, 5 (1999), 16731692
WEAKLYEINSTEIN HERMITIAN SURFACES
by V. APOSTOLOV and O. MUSKAROV
1. Introduction.
A Hermitian surface (M, J, h) is a complex surface (M, J) endowed
with a Jinvariant Riemannian metric h. If the K¨ahler form F(., .) = h(J., .)
of (M, J, h) is closed we obtain a K¨ahler surface. The Riemannian metric
h is said to be Einstein if its Ricci tensor Ric is a constant multiple of the
metric, i.e., if Ric = h, where the constant 4 is the scalar curvature of h.
Many efforts have been done to study compact Einstein Hermitian surfaces
(which, in general, give examples of nonhomogeneous Einstein 4spaces
[10], [8]). The compact K¨ahlerEinstein surfaces have been described by
completely resolving the corresponding complex MongeAmp`ere equations,
see [33], [3], [25], [28], [26], while the only known example of a compact, non
K¨ahler, Einstein Hermitian surface is the Hirzebruch surface F1
= CP2 ¯CP
2
with the Page metric [21]. Recently C. LeBrun [19] has proved that the
only other compact complex surfaces that could admit nonK¨ahler Einstein
