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Volume minimization and conformally Kähler, Einstein-Maxwell geometry. (English) Zbl 1410.53071
In this technical article, the authors construct so-called conformally Kähler Einstein-Maxwell metrics on compact manifolds by finding the critical points of a certain volume functional.
More precisely, let \((M,g)\) be a compact Kähler manifold with its induced almost complex tensor field \(J\). Then, a \(J\)-Hermitian manifold \((M,\tilde{g})\) is called a conformally Kähler Einstein-Maxwell (cKEM) manifold if (i) there exists a positive smooth function \(f\) on \(M\) such that \(g=f^2\tilde{g}\); (ii) the Hamiltonian vector field \(K:=J\mathrm{grad}_gf\) is Killing for both \(g\) and \(\tilde{g}\); (iii) the scalar curvature of \(\tilde{g}\) is constant. (Regarding terminology, one can demonstrate that in four dimensions, out of a cKEM manifold one can obtain a solution of the Riemannian Einstein-Maxwell equations.) Let \(G\subset \mathrm{Aut}_r(M,g)\) be a compact subgroup of the group of reduced automorphisms of \((M,g)\), with Lie algebra \({\mathfrak g}\). Then, to every pair \((K,a)\in {\mathfrak g}\times{\mathbb R}\) one can construct a certain metric \(\tilde{g}_{K,a}\) on \(M\) and an associated functional on \({\mathfrak g}\times{\mathbb R}\) given by \((K,a)\mapsto \mathrm{Vol}(M, \tilde{g}_{K,a})\). The main result of the paper is that cKEM metrics are critical points of this functional and a point \((K,a)\) is a critical point if and only if its so-called cKEM-Futaki invariant vanishes (see Theorem 1.1 and Section 1 for further explanations).
Section 2 contains the proof of the main result while Sections 3 and 4 are devoted to explicit examples of non-Kähler cKEM metrics and some computations with them.

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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