Surveys in Differential Geometry

Volume 18 (2013)

Calabi energies of extremal toric surfaces

Pages: 195 – 226

DOI: http://dx.doi.org/10.4310/SDG.2013.v18.n1.a5

Author

Claude Lebrun (Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York, U.S.A.)

Abstract

We derive a formula for the $L^2$ norm of the scalar curvature of any extremal Kähler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction of of Bach-flat metrics on appropriate 4-manifolds.

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