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

Title & Authors
THE FIRST POSITIVE EIGENVALUE OF THE DIRAC OPERATOR ON 3-DIMENSIONAL SASAKIAN MANIFOLDS
Kim, Eui Chul;

Abstract
Let ($\small{M^3}$, $\small{g}$) be a 3-dimensional closed Sasakian spin manifold. Let $\small{S_{min}}$ denote the minimum of the scalar curvature of ($\small{M^3}$, $\small{g}$). Let $\small{{\lambda}^+_1}$ > 0 be the first positive eigenvalue of the Dirac operator of ($\small{M^3}$, $\small{g}$). We proved in [13] that if $\small{{\lambda}^+_1}$ belongs to the interval $\small{{\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})}$, then $\small{{\lambda}^+_1}$ satisfies $\small{{\lambda}^+_1{\geq}{\frac{S_{min}+6}{8}}}$. In this paper, we remove the restriction "if $\small{{\lambda}^+_1}$ belongs to the interval $\small{{\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})}$" and prove $\small{{\lambda}^+_1{\geq}\;\{\frac{S_{min}+6}{8}\;for\;-\frac{3}{2}}$$\small{&}$$\small{lt;S_{min}{\leq}30, \\{\frac{1+\sqrt{2S_{min}}+4}{2}}\;for\;S_{min}{\geq}30}$.
Keywords
Dirac operator;eigenvalues;Sasakian manifolds;
Language
English
Cited by
1.
Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds, Annals of Global Analysis and Geometry, 2014, 45, 1, 67
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