Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics
type: Item
instance of: scholarly article
title: Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics
zbMATH Open document ID: 0889.53048
author: Andrew D. Hwang
author: Santiago R. Simanca
published in: Mathematische Annalen
publication date: 1997-09-09T00:00:00Z
review text: 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 energ…
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32M12
zbMATH DE Number: 1058820
zbMATH Keywords: extremal Kähler metric
zbMATH Keywords: Einstein metric
zbMATH Keywords: Hirzebruch surface
zbMATH Keywords: Calabi energy
zbMATH Keywords: potential energy functionals
reviewed by: Christos Baikoussis
MaRDI profile type: Publication
full work available at URL: s002080050104
OpenAlex ID: W2011122178
DOI: 10.1007/S002080050104
Recommended article: Conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces
Recommended article: Locally conformally Kähler metrics on certain non-Kählerian surfaces
Recommended article: Q4218992
Recommended article: Locally conformally Kähler metrics on Kato surfaces
Recommended article: Globally conformally Kähler Einstein metrics on certain holomorphic bundles
Recommended article: Locally conformally Kähler metrics on Hopf surfaces
Recommended article: A remark on extremal Kähler metrics
Recommended article: Distinguished Kahler Metrics on Hirzebruch Surfaces
Recommended article: Q6164781
Recommended article: Extremal K\"ahler metrics
description: scientific article; zbMATH DE number 1058820