×

On the Calabi energy of extremal Kähler metrics. (English) Zbl 0846.53048

Let \((M, \Omega)\) be a compact complex manifold of complex dimension \(n\) with a distinguished Kähler class. Let \(V_\Omega\) and \(S_\Omega\) be the volume and total scalar curvature of a metric \(g\). The main purpose of this article is to prove the following theorem concerning the Calabi energy. Theorem. Let \((M, \Omega)\) be fixed. There exists a number \(E_\Omega \geq S^2_\Omega/V_\Omega\) such that \(\int_Ms^2_g d \text{vol}_g \geq E_\Omega\) for all metrics \(g\) in \(\Omega\), with equality if and only if \(g\) is extremal. Thus the author answers affirmatively two questions of E. Calabi [Extremal Kähler metrics. II. Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 95-114 (1985; Zbl 0574.58006)].
Reviewer: N.Bokan (Beograd)

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds
32Q20 Kähler-Einstein manifolds
58E11 Critical metrics

Citations:

Zbl 0574.58006
PDFBibTeX XMLCite
Full Text: DOI