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Extrinsic estimates for eigenvalues of the Dirac operator

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Abstract

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Daguang Chen

  2. Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Daguang Chen

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  1. Daguang Chen
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Correspondence to Daguang Chen.

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Chen, D. Extrinsic estimates for eigenvalues of the Dirac operator. Math. Z. 262, 349–361 (2009). https://doi.org/10.1007/s00209-008-0376-8

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  • DOI: https://doi.org/10.1007/s00209-008-0376-8

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