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The Yau–Tian–Donaldson Conjecture

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Test Configurations, Stabilities and Canonical Kähler Metrics

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

In this chapter, we discuss the Yau–Tian–Donaldson conjecture from a historical point of view.

  • In Sect. 6.1, we briefly discuss the Calabi conjecture. The unsolved case of the Calabi conjecture motivates the Yau–Tian–Donaldson conjecture in Kähler–Einstein cases.

  • As mentioned in Sect. 6.2, the Yau–Tian–Donaldson conjecture in Kähler–Einstein cases was solved affirmatively by Chen, Donaldson and Sun and by Tian.

  • In Sect. 6.3, we define K-energy and modified K-energy for compact Kähler manifolds. This concept allows us to state the recent results of Chen and Cheng and of He on the existence of CSC Kähler metrics and extremal Kähler metrics.

  • Finally, in Sect. 6.4, various versions of the Yau–Tian–Donaldson conjecture will be considered in extremal Kähler cases.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, Graduate School of Science, Toyonaka, Osaka, Japan

    Toshiki Mabuchi

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  1. Toshiki Mabuchi
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Mabuchi, T. (2021). The Yau–Tian–Donaldson Conjecture. In: Test Configurations, Stabilities and Canonical Kähler Metrics. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-16-0500-0_6

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