Heat kernel asymptotics for Kohn Laplacians on CR manifolds

TL;DR

This paper derives heat kernel asymptotics for Kohn Laplacians on CR manifolds, providing new proofs of Morse inequalities and extending results to manifolds with transversal CR $ ext{R}$-actions.

Contribution

It establishes heat kernel asymptotics for Kohn Laplacians on CR manifolds under condition Y(q), including equivariant cases, and applies these to Morse inequalities.

Findings

Asymptotic formulas for heat kernels of Kohn Laplacians on CR manifolds.

Proofs of Morse inequalities using heat kernel methods.

Extension of results to manifolds with transversal CR $ ext{R}$-actions.

Abstract

Let $X$ be an abstract orientable not necessarily compact CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ 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 $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an application, we get $\mathbb R$-equivariant Morse inequalities on compact CR manifolds with transversal CR $\mathbb R$-action.