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

摘要

Let $( M^3 , g)$ be a 3-dimensional closed Sasakian spin manifold. Let $S_{m min}$ 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 $lambda_1^{+} in left( rac{1}{2}, rac{5}{2} ight)$, then $lambda_1^{+}$ satisfies $ lambda_1^{+} geq rac{S_{m min} + 6}{8}$. In this paper, we remove the restriction ``if $lambda_1^{+}$ belongs to the interval $ lambda_1^{+} in ( rac{1}{2} , rac{5}{2} )$" and prove [ lambda_1^{+} geq left{ egin{array}{ll} rac{ S_{m min} +6}{8} & hbox{ for } - rac{3}{2} < S_{m min} leq 30 , rac{1 + sqrt{2 , S_{m min} + 4}}{2} & hbox{ for} S_{m min} geq 30 . end{array} ight. ]