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Extremal Kähler metrics on minimal ruled surfaces. (English) Zbl 0921.53033
Assume that a certain real cohomology class on a compact complex manifold \(M\) can be represented by a Kähler 2-form \(\omega\). One can consider the set \([\omega ]^{+}\) of all Kähler forms which represent the same cohomology class and the associated Kähler metric for each of these forms. E. Calabi suggested in his paper [Ann. Math. Stud. 102, 259-290 (1982; Zbl 0487.53057)] to consider extrema of the functional given by the square of the \(L^2\)-norm of the scalar curvature with the functional restricted to the set of Kähler metrics associated to the forms in \([\omega ]^{+}\). Such metrics are called extremal Kähler metrics and are the object of the present study. Kähler metrics of constant scalar curvature are examples of extremal metrics, but not every extremal metric has constant scalar curvature.
The author constructs an interesting family of extremal Kähler metrics on certain minimal ruled complex surfaces, called pseudo-Hirzebruch surfaces (ruled over Riemann surfaces of genus greater than one). These metrics have nonconstant scalar curvature. Moreover, the author shows that if every Kähler class on a surface of this type has an extremal Kähler metric, then there exist some Kähler classes on the surface for which the extremal representative fails to be unique.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E11 Critical metrics
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