Abstract
We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.
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Acknowledgements
We owe many thanks to Bo Guan, Bo Yang, Adrian Lam, and Fangyang Zheng for their insights and help in deriving these results. Finally, thanks are due to Kori Brady and Fangyang Zheng for their edits and help in making the writing more clear.
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Khan, G.J.H. Eigenvalues of the complex Laplacian on compact non-Kähler manifolds. Ann Glob Anal Geom 53, 233–249 (2018). https://doi.org/10.1007/s10455-017-9574-0
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DOI: https://doi.org/10.1007/s10455-017-9574-0