Strongly extremal Kähler metrics
The author considers a compact complex manifold of the Kähler type, introduces a curvature functional and searches for its critical points. The author, successively, derives a formula that relates the Kähler and Ricci forms of such metrics on surfaces. An important result obtained is the following: if the metrics have a nonzero constant scalar curvature, then they must be Einstein. In the case of complex surfaces the metrics proposed (if the scalar curvature is nonconstant) are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. The Hessian of the lower bound of the fundamental at a critical extremal class is also calculated, showing that in low dimensions these classes are weakly stable minima for the said bound. Starting from important results contained in \textit{C. LeBrun} [Lect. Notes Pure Appl. Math. 184, 167-176 (1997; Zbl 0876.53024)] and \textit{C. LeBrun} and \textit{S. R. Simanca} [Geom. Funct. Anal. 4, 298-336 (1994; Zbl 0801.53050)] the author introduces some applications concerning the two-point blow-up of \(\mathbb{C} \mathbb{P}^2\).
scientific article; zbMATH DE number 1422971