The first positive eigenvalue of the Dirac operator on 3-dimensional Sasakian manifolds (Q2856982)

Closed Riemannian spin manifolds \(M^n\) of dimension \(n \equiv 3\mod 4\) have a Dirac operator \(D\) with `non-symmetric' spectrum, typically. Hence it is important to estimate the first positive eigenvalue \(\lambda_1^+ > 0\) from below and the first negative eigenvalue \(\lambda_1^- <0\) from above, and try to interpret the limiting case, for example in terms of the scalar curvature, in the spirit of \textit{T. Friedrich}'s inequality [Math. Nachr. 97, 117--146 (1980; Zbl 0462.53027)].NEWLINENEWLINENEWLINEThe paper reviewed discusses the problem for (closed) \(3\)-dimensional (spin) Sasaki manifolds \(M\). Assuming the scalar curvature is bounded, \(S_{\min}>-2\), the author bounds the eigenvalues \(\lambda\) of \(D\):NEWLINENEWLINEif \(\lambda<1/2\), then \(\lambda \leq \tfrac 12 (1-\sqrt{2S_{\min}+4})\), NEWLINENEWLINENEWLINEif \(1/2<\lambda<9/2\), then \(\lambda \geq \tfrac 18 (6+S_{\min})\), NEWLINENEWLINENEWLINEif \(\lambda\geq 9/2\), then \(\lambda \geq \tfrac 12 (1+\sqrt{2S_{\min}+4})\). NEWLINENEWLINENEWLINEIn all cases, \(\lambda\) equals the extreme value if and only if the scalar curvature is constant and there exists an \(\eta\)-Killing spinor (for this notion see [the author and \textit{T. Friedrich}, J. Geom. Phys. 33, No. 1--2, 128--172 (2000; Zbl 0961.53023)]). As a consequence, one can estimate \(\lambda_1^+\) as follows:NEWLINENEWLINEif \(S_{\min}\in (-3/2, 30]\), then \(\lambda_1^+\geq \tfrac 18 (6+S_{\min})\), NEWLINENEWLINENEWLINEif \(S_{\min}\in [30,\infty)\), then \(\lambda_1^+\geq \tfrac 12 (1+\sqrt{2S_{\min}+4})\).NEWLINENEWLINEAgain, constant scalar curvature and an \(\eta\)-Killing spinor characterise the minimum values.