Ju l 2 01 9 Eigenvalue estimates via Hömander ’ s L 2-method

In this paper, we apply Hömander’s weighted L-method([9]) to study eigenvalues of Dirac operators of Dirac bundles over Riemannian manifolds. The conformal covariance([13]) of the classical Dirac operators played an important role in estimating eigenvalues of the classical Dirac operator on spin manifolds, see [9]-[12], [8], [5] and the references therein. Different from spinor bundles over spin manifolds, the connection of the Dirac bundle is not determined by the Levi-Civita connection of the underlying manifold. In general, we don’t have conformal covariance for Dirac operators of Dirac bundles. Bär([2]) generalized the Hijazi estimate([9]) to Dirac operators of Dirac bundles over closed manifolds. To avoid the use of conformal covariance, the modified connection is the key technique in Bär’s proof. We will consider eigenvalue bounds for Dirac operators of Dirac bundles under both local(Definitions 3.1, 3.2) and global(Definitions 3.5, 3.7) boundary conditions. Our weighted L-identity is given by Lemma 2.5 below where the boundary terms will be dealt with using the Morrey trick for the case of local boundary condition and the boundary Dirac operator for the case of global condition. By a rescaling argument, we also obtain lower bounds in terms of the volume of the underlying manifolds where the Li-Zhu inequality ([17]) is the fundamental tool. Recently, Chang, Chen and Wu([4]) also study eigenvalue estimates in CR geometry by establishing weighted Rayleigh formula, their method is closely related to this paper.