Abstract
We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR N. From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS N or,ifM is complex, a holomorphic isometric immersion intoPC N.
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Bunke, U. Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions. Ann Glob Anal Geom 9, 109–116 (1991). https://doi.org/10.1007/BF00776850
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DOI: https://doi.org/10.1007/BF00776850