Toric extremal Kähler-Ricci solitons are Kähler-Einstein. (English) Zbl 1381.53076
The authors prove that Calabi extremal Kähler-Ricci solitons on compact toric Kähler manifolds are Kähler-Einstein, see Theorem 2.
Extremal Kähler metrics are critical points of the Calabi functional, that is, the \(L^2\)-norm of the scalar curvature. Kähler-Ricci solitons are Kähler metrics \(\omega\) satisfying \(\rho+c\omega=\mathcal{L}_X\omega\), where \(\rho\) is the Ricci form, \(c\) is a real constant, and \(X\) is a holomorphic vector field. Toric manifolds are compact Kähler \(2n\)-dimensional manifolds admitting an effective Hamiltonian action of an \(n\)-dimensional torus by Kähler automorphisms.
In [Proc. Am. Math. Soc. 144, No. 2, 813–821 (2016; Zbl 1337.53053)], the authors proved that compact extremal Kähler-Ricci solitons with positive holomorphic sectional curvature are Kähler-Einstein. Note that toric Kähler manifolds can have holomorphic sectional curvature of any sign. More in general, the authors ask whether it is true that every extremal Kähler-Ricci solitons are Einstein, see Problem 2.
Extremal Kähler metrics are critical points of the Calabi functional, that is, the \(L^2\)-norm of the scalar curvature. Kähler-Ricci solitons are Kähler metrics \(\omega\) satisfying \(\rho+c\omega=\mathcal{L}_X\omega\), where \(\rho\) is the Ricci form, \(c\) is a real constant, and \(X\) is a holomorphic vector field. Toric manifolds are compact Kähler \(2n\)-dimensional manifolds admitting an effective Hamiltonian action of an \(n\)-dimensional torus by Kähler automorphisms.
In [Proc. Am. Math. Soc. 144, No. 2, 813–821 (2016; Zbl 1337.53053)], the authors proved that compact extremal Kähler-Ricci solitons with positive holomorphic sectional curvature are Kähler-Einstein. Note that toric Kähler manifolds can have holomorphic sectional curvature of any sign. More in general, the authors ask whether it is true that every extremal Kähler-Ricci solitons are Einstein, see Problem 2.
Reviewer: Daniele Angella (Firenze)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 58D19 | Group actions and symmetry properties |
Citations:
Zbl 1337.53053References:
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