THE FIRST POSITIVE EIGENVALUE OF THE DIRAC OPERATOR ON 3-DIMENSIONAL SASAKIAN MANIFOLDS

Let ($M^3$, $g$) be a 3-dimensional closed Sasakian spin manifold. Let $S_{min}$ 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 ${\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})$, then ${\lambda}^+_1$ satisfies ${\lambda}^+_1{\geq}{\frac{S_{min}+6}{8}}$. 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 $${\lambda}^+_1{\geq}\;\{\frac{S_{min}+6}{8}\;for\;-\frac{3}{2}<S_{min}{\leq}30, \\{\frac{1+\sqrt{2S_{min}}+4}{2}}\;for\;S_{min}{\geq}30$$.