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Gradient and Eigenvalue Estimates on the Canonical Bundle of Kähler Manifolds

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Abstract

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kähler manifolds, where m is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of (m, 0) forms, which involves only the Ricci curvature and the gradient of the scalar curvature.

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Acknowledgements

We wish to thank Professor Jiaping Wang for very helpful suggestions on Lemma 2.4. Z. L. is supported by NSF Grant DMS-19-08513. Q. Z. is supported by the Simons Foundation Grant 710364. M. Z. is supported by Shanghai Science and Technology Innovation Program Basic Research Project of STCSM20JC1412900, Science and Technology Commission of Shanghai Municipality (STCSM) No. 18dz2271000, and NSFC No.11971168.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Zhiqin Lu

  2. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    Qi S. Zhang

  3. School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

    Meng Zhu

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  1. Zhiqin Lu
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  2. Qi S. Zhang
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Correspondence to Zhiqin Lu.

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Lu, Z., Zhang, Q.S. & Zhu, M. Gradient and Eigenvalue Estimates on the Canonical Bundle of Kähler Manifolds. J Geom Anal 31, 10304–10335 (2021). https://doi.org/10.1007/s12220-021-00647-8

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  • DOI: https://doi.org/10.1007/s12220-021-00647-8

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