An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds
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 \(\| df\|_{\infty}=\max | df_ x|.\)
scientific article; zbMATH DE number 4190615