Abstract
Let \((X,\omega _0)\) be a compact complex manifold of complex dimension n endowed with a Hermitian metric \(\omega _0\). The Chern–Yamabe problem is to find a conformal metric of \(\omega _0\) such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of \((X,\omega _0)\). On the other hand, we prove a version of conformal Schwarz lemma on \((X,\omega _0)\). All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.
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Acknowledgements
The authors would like to thank the referee for his/her valuable comments which helped to improve the manuscript. The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019041021), and by Korea Institute for Advanced Study (KIAS) grant funded by the Korea government (MSIP).
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Ho, P.T. Results Related to the Chern–Yamabe Flow. J Geom Anal 31, 187–220 (2021). https://doi.org/10.1007/s12220-019-00255-7
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DOI: https://doi.org/10.1007/s12220-019-00255-7