Bulletin of the
Korean Mathematical Society BKMS

Let $( M^3 , g)$ be a 3-dimensional closed Sasakian spin manifold. Let $S_{\rm min}$ denote the minimum of the scalar curvature of $(M^3, g)$. Let $\lambda_1^{+} > 0$ be the first positive eigenvalue of the Dirac operator of $(M^3, g)$. We proved in [13] that if $\lambda_1^+$ belongs to the interval $\lambda_1^{+} \in \left( \frac{1}{2}, \frac{5}{2} \right)$, then $\lambda_1^{+}$ satisfies $ \lambda_1^{+} \geq \frac{S_{\rm min} + 6}{8}$. In this paper, we remove the restriction ``if $\lambda_1^{+}$ belongs to the interval $ \lambda_1^{+} \in ( \frac{1}{2} , \frac{5}{2} )$" and prove \[ \lambda_1^{+} \ \geq \ \left\{ \begin{array}{ll} \frac{ S_{\rm min} +6}{8} \ & \hbox{ for } \ - \frac{3}{2} < S_{\rm min} \leq 30 , \\ \frac{1 + \sqrt{2 \, S_{\rm min} + 4}}{2} \ & \hbox{ for} \ \ \ S_{\rm min} \geq 30 . \end{array} \right. \]

Keywords: Dirac operator, eigenvalues, Sasakian manifolds

MSC numbers: 53C25, 53C27