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

  • Kim, Eui Chul
  • Received : 2011.09.19
  • Published : 2013.03.31

Abstract

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 $${\lambda}^+_1{\geq}\;\{\frac{S_{min}+6}{8}\;for\;-\frac{3}{2}<S_{min}{\leq}30, \\{\frac{1+\sqrt{2S_{min}}+4}{2}}\;for\;S_{min}{\geq}30$$.

Keywords

Dirac operator;eigenvalues;Sasakian manifolds

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Cited by

  1. Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds vol.45, pp.1, 2014, https://doi.org/10.1007/s10455-013-9388-7

Acknowledgement

Supported by : Andong National University