Geodesic-Einstein metrics and nonlinear stabilities

Huitao Feng Kefeng Liu Xueyuan Wan

Differential Geometry mathscidoc:1912.43087

Transactions of the American Mathematical Society, 371, (11), 8029-8049, 2019
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.
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  title={Geodesic-Einstein metrics and nonlinear stabilities},
  author={Huitao Feng, Kefeng Liu, and Xueyuan Wan},
  booktitle={Transactions of the American Mathematical Society},
Huitao Feng, Kefeng Liu, and Xueyuan Wan. Geodesic-Einstein metrics and nonlinear stabilities. 2019. Vol. 371. In Transactions of the American Mathematical Society. pp.8029-8049.
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