Hessian of the Ricci-Calabi functional
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- by Satoshi Nakamura PDF
- Proc. Amer. Math. Soc. 147 (2019), 1247-1254 Request permission
Abstract:
The Ricci-Calabi functional is a functional on the space of Kähler metrics of Fano manifolds. Its critical points are called generalized Kähler-Einstein metrics. In this article, we show that the Hessian of the Ricci-Calabi functional is non-negative at generalized Kähler-Einstein metrics. As its application, we give another proof of a Matsushima type decomposition theorem for holomorphic vector fields, which was originally proved by Mabuchi. We also discuss a relation to the inverse Monge-Ampère flow developed recently by Collins-Hisamoto-Takahashi.References
- Eugenio Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95–114. MR 780039
- T. Collins, T. Hisamoto, and R. Takahashi, The inverse Monge-Ampère flow and application to Kähler-Einstein metrics, preprint, 2017, arXiv:1712.01685v1.
- S. Donaldson, The Ding functional, Berndtsson convexity and moment maps, in Geometry, Analysis and Probability, Progr. Math., Birkhauser/Springer, 310 2017, pp. 57–67.
- F. T. Fong, Boltzmann’s entropy and Kähler-Ricci solitons, preprint, 2016, arXiv:1605.08019v1.
- Akito Futaki, Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics, vol. 1314, Springer-Verlag, Berlin, 1988. MR 947341, DOI 10.1007/BFb0078084
- Akito Futaki, Holomorphic vector fields and perturbed extremal Kähler metrics, J. Symplectic Geom. 6 (2008), no. 2, 127–138. MR 2434437
- Akito Futaki and Toshiki Mabuchi, Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann. 301 (1995), no. 2, 199–210. MR 1314584, DOI 10.1007/BF01446626
- A. Futaki and H. Ono, Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group, preprint, 2017, arXiv:1708.01958v2.
- P. Gauduchon, Calabis extremal metrics: An elementary introduction, lecture notes, http://germanio.math.unifi.it/wp-content/uploads/2015/03/dercalabi.pdf
- Y. Li and B. Zhou, Mabuchi metrics and properness of the modified Ding functional, preprint, 2017, arXiv:1709.03029.
- Toshiki Mabuchi, Kähler-Einstein metrics for manifolds with nonvanishing Futaki character, Tohoku Math. J. (2) 53 (2001), no. 2, 171–182. MR 1829977, DOI 10.2748/tmj/1178207477
- Yozô Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J. 11 (1957), 145–150 (French). MR 94478
- S. Nakamura, Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds, to appear in Tohoku Math. J., preprint, 2017, arXiv:1706.01608v4.
- S. Nakamura, Remarks on modified Ding functional for toric Fano manifolds, preprint, 2017, arXiv:1710.06828.
- Y. Nitta, S. Saito, and N. Yotsutani, Relative GIT stabilities of toric Fano manifolds in low dimensions, preprint, 2017, arXiv:1712.01131v1.
- Lijing Wang, Hessians of the Calabi functional and the norm function, Ann. Global Anal. Geom. 29 (2006), no. 2, 187–196. MR 2249564, DOI 10.1007/s10455-005-9014-4
- Y. Yao, Mabuchi metrics and relative Ding stability of toric Fano varieties, preprint, 2017, arXiv:1701.04016v2.
Additional Information
- Satoshi Nakamura
- Affiliation: Mathematical Institute, Tohoku University, Sendai, 980-8578 Japan
- Email: satoshi.nakamura.e6@tohoku.ac.jp
- Received by editor(s): January 10, 2018
- Received by editor(s) in revised form: July 9, 2018
- Published electronically: December 6, 2018
- Additional Notes: The author was partly supported by Grant-in-Aid for JSPS Fellowships for Young Scientists, No. 17J02783.
- Communicated by: Jia-Ping Wang
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1247-1254
- MSC (2010): Primary 53C25; Secondary 53C55, 58E11
- DOI: https://doi.org/10.1090/proc/14321
- MathSciNet review: 3896070