Lower bounds for eigenvalues of the Dirac operator on surfaces of rotation. (English) Zbl 0976.53050
The author studies the eigenvalues of the Dirac operator on a two-dimensional sphere equipped with a Riemannian metric that is invariant under a free circle action. Let \(f_{\max}\) be the maximal length of an orbit. Then she proves the theorem:
Any eigenvalue \(\lambda\) of the Dirac operator satisfies \(|\lambda |\geq {1\over 2 f_{\max}}\).
The multiplicity of an eigenvalue \(\lambda_n\), \(n>0\) with \({2n-1\over 2f_{\max}}\leq \lambda_n \leq {2n+1\over 2f_{\max}}\) is at most \(2n\).
Any eigenvalue \(\lambda\) of the Dirac operator satisfies \(|\lambda |\geq {1\over 2 f_{\max}}\).
The multiplicity of an eigenvalue \(\lambda_n\), \(n>0\) with \({2n-1\over 2f_{\max}}\leq \lambda_n \leq {2n+1\over 2f_{\max}}\) is at most \(2n\).
Reviewer: Bernd Ammann (Hamburg)
MSC:
| 53C27 | Spin and Spin\({}^c\) geometry |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
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