Heat kernel asymptotics for Kohn Laplacians on CR manifolds

Abstract

Let X be an abstract orientable (not necessarily compact) CR manifold of dimension 2n+1, n≥1, and let Lk be the k-th tensor power of a CR complex line bundle L over X. Suppose that condition Y(q) holds at each point of X, we establish asymptotics of the heat kernel of Kohn Laplacian with values in Lk. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When X admits a transversal CR R-action, we also establish asymptotics of the R-equivariant heat kernel of Kohn Laplacian with values in Lk. As an application, we get R-equivariant Morse inequalities on compact CR manifolds with transversal CR R-action.