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Existence of coupled Kähler–Einstein metrics using the continuity method

Vamsi Pritham Pingali

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

https://doi.org/10.1142/S0129167X18500416Cited by:7 (Source: Crossref)

Abstract

In this paper, we prove the existence of coupled Kähler–Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual Kähler–Einstein metrics, we reduce existence in the Fano case to a $C^{0}$ estimate.

Keywords:

AMSC: 32Q20, 53C25

References

1. Z. Blocki, On the uniform estimate in the Calabi-Yau theorem, Sci. China, Ser. A 48 (2005) 244–247. Crossref, Web of Science, Google Scholar
2. X. Chen, S. Donaldson and S. Sun, Kähler–Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc. 28(1) (2015) 183–197. Crossref, Web of Science, Google Scholar
3. X. Chen, S. Donaldson and S. Sun, Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than $2 π$ , J. Am. Math. Soc. 28(1) (2015) 199–234. Crossref, Web of Science, Google Scholar
4. X. Chen, S. Donaldson and S. Sun, Kähler–Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2 π$ and completion of the main proof, J. Am. Math. Soc. 28(1) (2015) 235–278. Crossref, Web of Science, Google Scholar
5. X. X. Chen and W. He, The complex Monge–Ampére equation on compact Kähler manifolds, Math. Ann. 354(4) (2012) 1583–1600. Crossref, Web of Science, Google Scholar
6. V. Datar and G. Szḱelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016) 975–1010. Crossref, Web of Science, Google Scholar
7. S. F. Hassan and R. A. Rosen, Bimetric gravity from ghost-free massive gravity, J. High Energy Phys. 2012(2) (2012) 1–12. Google Scholar
8. J. Hultgren and D. W. Nyström, Coupled Kähler–Einstein metrics, to appear in Int. Math. Res. Not., arXiv:1608.07209. Google Scholar
9. S. Kolodziej, The complex Monge-Ampère equation, Acta. Math. 180 (1998) 69–117. Crossref, Web of Science, Google Scholar
10. G. Tian, Canonical Metrics in Kähler Geometry (Birkhäuser, 2012). Google Scholar
11. Y. Wang, On the $C^{2, α}$ -regularity of the complex Monge-Ampére equation, Math. Res. Lett. 19(4) (2012) 939–946. Crossref, Web of Science, Google Scholar
12. S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex monge-ampère equation I, Comm. Pure App. Math. 31(3) (1978) 339–411. Crossref, Web of Science, Google Scholar