Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics
Let \((M, J)\) be a compact complex manifold of Kähler type, and \(\mathfrak {M}\) be the space of Kähler forms on \((M, J),\) namely \(\mathfrak {M}\) is an open cone in the infinite-dimensional affine space of closed (1,1)-forms. One way to single out canonical metrics on \((M, J)\) is to introduce a curvature functional and search for its critical metrics. A natural functional \(\Phi \) to consider is the square of the \(L^2\)-norm of the scalar curvature. The now standard procedure (initiated by \textit{E. Calabi} [`The space of Kähler metrics', Proc. Int. Congr. Math. Amsterdam 2, 206-207 (1954)]) is to fix a class \(\Omega \in H^{1,1}(M, {\mathbb R})\subset H^2(M, {\mathbb R})\) in the Kähler cone and to search for critical metrics of the restriction of \(\Phi \) to \({\mathfrak {M}}_{\Omega} =\{\omega \in {\mathfrak {M}}: [\omega ]= \Omega \}\). Such metrics are called extremal in the sense of Calabi. The Calabi energy of \(\Omega \) is the restricted functional \(\Phi_{\Omega}:= \Phi |_{{\mathfrak {M}}_{\Omega}}\). In this article the authors consider the Calabi energy and potential energy functionals and the critical points of these are studied for the Hirzebruch surfaces \( {\mathfrak {F}}_n= {\mathbb {P}} ({O_{\mathbb {P}^1}} (-n) \oplus O_{\mathbb{P}^1})\). The main results are the following: (i) For each \(n\geq 0, n\neq 2\), there is a Kähler class on \({\mathfrak {F}}_n \), unique up to scaling, which is a global minimum of the potential energy. On the surface \({\mathfrak {F}}_2 \), the minimizing class is degenerate. For each \(n\), the potential energy has no other stationary points. In particular, for each \(n\geq 0, n\neq 2\), there is a Kähler form on \({\mathfrak {F}}_n \), unique up to scaling and automorphisms, which is a global minimum of the functional \(\Phi \) among all Kähler forms on \({\mathfrak {F}}_n \). (ii) For each \(n\geq 1, n\neq 2\), the degenerate Kähler class on \({\mathfrak {F}}_n \) is a local maximum of the potential energy but is not stationary. (iii) Each of the extremal metrics in (i), and each degenerate extremal metric, is locally conformal to an Einstein metric except possibly along a real submanifold. On the Hirzebruch surfaces \({\mathfrak {F}}_n \), these are the only Kähler metrics with this property.
scientific article; zbMATH DE number 1058820