## Integral inequalities for holomorphic maps and applications

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## Abstract:

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the Kähler-Ricci flow. The equality case is characterized for some special settings.## References

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## Additional Information

**Yashan Zhang**- Affiliation: School of Mathematics and Hunan Province Key Lab of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha 410082, People’s Republic of China
- MR Author ID: 1234610
- Email: yashanzh@hnu.edu.cn
- Received by editor(s): November 1, 2019
- Received by editor(s) in revised form: August 11, 2020
- Published electronically: January 21, 2021
- Additional Notes: The author was partially supported by Fundamental Research Funds for the Central Universities (No. 531118010468) and National Natural Science Foundation of China (No. 12001179)
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2341-2358 - MSC (2020): Primary 53C55
- DOI: https://doi.org/10.1090/tran/8293
- MathSciNet review: 4223018