Estimates of small Dirac eigenvalues on 3-dimensional Sasakian manifolds

On a 3-dimensional closed Sasakian spin manifold (M3,g)(M3,g), the spectrum of the Dirac operator D   is in general not symmetric with respect to zero. Let λ1−<0 and λ1+>0 be the first negative and positive eigenvalue of D  , respectively. Let SminSmin denote the minimum of the scalar curvature of (M3,g)(M3,g) with Smin>−32. We prove in this paper that λ1−⩽1−2Smin+42 holds generally and that λ1+ satisfies λ1+⩾Smin+68 whenever λ1+ belongs to the interval λ1+∈(12,52). It turns out that each of these estimates improves Friedrich's inequality for the first eigenvalue of the Dirac operator [Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980) 117–146].