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On Calabi extremal Kähler-Ricci solitons. (English) Zbl 1337.53053

The authors study Kähler metrics which are both Calabi extremal and Kähler-Ricci solitons. Namely, for Kähler-Ricci solitons, they establish some characterizations of being extremal in terms of the length of the complex Hessian of its potential function and in terms of the Riemann curvature tensor. By means of these, they prove that any extremal Kähler-Ricci soliton with positive holomorphic sectional curvature is Einstein. Also, a condition is pointed out about the isometry group of a non-Einstein extremal Kähler-Ricci soliton.

MSC:

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