Lower bounds for eigenvalues of the Dirac operator on \(n\)-spheres with \(SO(n)\)-symmetry
Let \(g\) be a Riemannian metric on the sphere \(S^n\) admitting an isometric \(SO(n)\)-action. Hence the metric is of cohomogeneity 1. Write \(f_{\max}^{n-1}\cdot \text{vol}(S^{n-1})\) for the maximal volume of orbits. The main result of the paper is Theorem. The spectrum of the Dirac operator on \((S^n,g)\) is symmetric and the multiplicity of an eigenvalue \(\lambda\) with \({(n-1)/2 + k \over f_{\max}} \leq \lambda < {(n-1)/2 + k +1 \over f_{\max}}\), \(k\geq 0\), is at most \(2^{[{n \over 2}]}\binom {n-1+k}{k}\). In particular, the smallest positive eigenvalue satisfies \(\lambda \geq (n-1)/(2f_{\max})\) and has multiplicity at most \(2^{[{n \over 2}]}\). This extends work by the same author on the 2-dimensional case [cf. J. Geom. Phys. 31, No. 2-3, 209-216 (1999; Zbl 0976.53050), see the preceding review]. For the proof one uses the high symmetry of the metric to perform a separation of variables and reduces the problem to the study of a system of ordinary differential equations. This lower eigenvalue bound compares nicely with the following upper bound. Theorem [the reviewer, Ann. Global Anal. Geom. 16, No. 6, 573-596 (1998; Zbl 0921.58065)]. Let \(M\subset\mathbb{R}^{n+1}\) be a closed hypersurface with mean curvature \(H\). Then the Dirac operator of \(M\) has at least \(2^{[{n \over 2}]}\) eigenvalues \(\lambda\) satisfying \(\lambda^2 \leq {n^2 \int_M H^2 \over 4 \text{vol}(M)}\). Now look at the following example. Let \(M\) be the cylinder \([0,L]\times S^{n-1}\) with hemispheres glued to it at the ends. The upper bound gives \(2^{[{n \over 2}]}\) Dirac eigenvalues \(\lambda(L)\) with \[ \lambda(L)^2 \leq {(n-1)^2L \text{vol}(S^{n-1}) + n^2\text{vol}(S^n) \over 4(L \text{vol}(S^{n-1})+\text{vol}(S^n))} \to \left({n-1 \over 2}\right)^2 \] for \(L\to \infty\) while the new lower bound yields \(\lambda(L) \geq {n-1 \over 2}\) for all \(L\) with multiplicity at most \(2^{[{n \over 2}]}\).
scientific article; zbMATH DE number 1422966
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