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Extremal Kähler metrics. II. (English) Zbl 0574.58006

Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 95-114 (1985).
[For the entire collection see Zbl 0561.00010; for part I see Semin. differ. geom., Ann. Math. Stud. 102, 259-290 (1982; Zbl 0487.53057).]
From the author’s abstract: ”Given a compact, complex manifold M with a Kähler metric, we fix the deRham cohomology class \(\Omega\) of the Kähler metric, and consider the function space \({\mathcal G}_{\Omega}\) of all Kähler metrics in M in that class. To each (g)\(\in {\mathcal G}_{\Omega}\) we assign the non-negative real number \(\Phi (g)=\int_{M}R^ 2_ g dV_ g\) \((R_ g=scalar\) curvature, \(dV_ g=volume\) element). Aiming to find a (g)\(\in {\mathcal G}_{\Omega}\) that minimizes the function \(\Phi\), we study the geometric properties in M of any (g)\(\in {\mathcal G}_{\Omega}\) that is a critical point of \(\Phi\), with the following results:
1) Any metric (g) that is a critical point of \(\Phi\) is necessarily invariant under a maximal compact subgroup of the identity component \({\mathfrak h}_ 0(M)\) of the complex Lie group of all holomorphic automorphisms of M.
2) Any critical metric (g)\(\in {\mathcal G}_{\Omega}\) of \(\Phi\) achieves a local minimum value of \(\Phi\) in \({\mathcal G}_{\Omega}\); the component of (g) in the critical set of \(\Phi\) coincides with the orbit of (g) under the action of the group \({\mathfrak h}_ 0(M)\), it is diffeomorphic to an open euclidean ball, and the critical set is always non-degenerate in the sense of \({\mathfrak h}_ 0(M)\)-equivariant Morse theory.
3) If there exists a (g)\(\in {\mathcal G}_{\Omega}\) with constant scalar curvature R, then it achieves an absolute minimum value of \(\Phi\) ; furthermore every critical metric in \({\mathcal G}_{\Omega}\) has constant R, and achieves the same value of \(\Phi\).
4) Whenever the existence of a critical Kähler metric (g) can be guaranteed (i.e., always, according to a conjecture), then Futaki’s obstruction determines a necessary and sufficient condition for the existence of a (g)\(\in {\mathcal G}_{\Omega}\) with constant scalar curvature.”
Reviewer: J.Girbau

MSC:

58E11 Critical metrics
53B35 Local differential geometry of Hermitian and Kählerian structures
53C55 Global differential geometry of Hermitian and Kählerian manifolds