The first eigenvalue of the Kohn–Dirac operator on CR manifolds

Abstract

On pseudo-Hermitian spin manifolds the Webster–Tanaka connection gives rise to a first order, sub-elliptic differential operator Dθ on spinors, the so-called Kohn–Dirac operator. This operator is formally self-adjoint and splits into a sum of CR-covariant operators D+ and D−. A Schrödinger–Lichnerowicz-type formula can be established, which gives rise to lower estimates for the first non-zero eigenvalue of Dθ on arbitrary spinors over closed manifolds. In the limiting case we discuss a Ψ-Killing spinor equation. On pseudo-Einstein spaces without torsion of positive Webster scalar curvature we construct such Ψ-Killing spinors. Their existence is related to 3-Sasakian geometry.