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Existence of weak conical Kähler–Einstein metrics along smooth hypersurfaces

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Abstract

The existence of weak conical Kähler–Einstein metrics along smooth hypersurfaces with cone angle between \(0\) and \(2\pi \) is obtained by studying a family of Aubin’s (J Funct Anal 57:143–153, 1984) continuity paths and obtaining a uniform \(C^2\) estimate by a local Moser’s iteration technique. As soon as the \(C^0\) estimate is achieved, the local Moser’s iteration technique could improve the rough \(C^2\) estimate in Chen et al. (J Am Math Soc 28:183–197, 2015) to a uniform \(C^2\) estimate. Since in the cases of negative and zero Ricci curvature, the \(C^0\) estimate is unobstructed, the weak conical Kähler–Einstein metrics are obtained; while in the case of positive Ricci curvature, the \(C^0\) estimate is achieved under the assumption of the properness of the Twisted K-Energy. The method used in this paper does not depend on the bound of the holomorphic bisectional curvature of any global background conical Kähler metrics.

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Notes

  1. The using of Chern–Lu inequality in the Kähler–Einstein context first appeared in the preprint of Jeffres et al. [14] for the conical singular version, we are following Chen et al. [7] for the smooth version.

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Acknowledgments

The would like to express great thanks to his advisor, Professor Xiuxiong Chen, who gave him constant encouragement and showed great patience during working on this problem. The gratitude also goes to Dr. Song Sun, Dr. Haozhao Li and Dr. Yuanqi Wang, Yu Zeng and Long Li for helpful discussion and providing useful notes to me at several points. The communication with Dr. Henri Guenancia also helped me clarify several things which were not clear to me before. Finally, the author would also ilke to thank the anonymous referee for suggestions on the reorganization of this article.

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  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Chengjian Yao

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  1. Chengjian Yao
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Yao, C. Existence of weak conical Kähler–Einstein metrics along smooth hypersurfaces. Math. Ann. 362, 1287–1304 (2015). https://doi.org/10.1007/s00208-014-1140-5

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