Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds

In this paper, we establish a new eigenvalue estimate for the Kohn-Dirac operator on a compact CR manifold. The equality case of this estimate is characterized by the existence of a CR twistor spinor. We then classify CR manifolds carrying such spinors by showing that the Webster Ricci tensor has at most two eigenvalues. In this context, we construct several examples.