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An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds. (English) Zbl 0722.53036

Let \((M^ n,g)\) be a compact Riemannian spin manifold. The author proves several upper bounds depending on the sectional curvature and the injectivity radius for the first eigenvalue of the Dirac operator. These results are obtained by a comparison argument with the sphere. Let \(0=\lambda^ 2_ 0<\lambda^ 2_ 1<\lambda^ 2_ 2..\) denote the eigenvalues of \(D^ 2\) and let \(m_ j\) be the multiplicity of \(\lambda^ 2_ j\). Consider a smooth map \(f: M^{2m}\to S^{2m}\) with degree \(\deg (f)\geq 2^{m-1}(m_ 0+...+m_{k-1})+1.\) Then the k-th eigenvalue of the Dirac operator is bounded by \(| \lambda_ k| \leq 2^{m-1}\sqrt{m}\| df\|_{\infty}\) where \(\| df\|_{\infty}=\max | df_ x|.\)

MSC:

53C20 Global Riemannian geometry, including pinching
58J05 Elliptic equations on manifolds, general theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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