Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds

TL;DR

This paper solves the tangential Kohn Laplacian on certain non-compact CR manifolds by leveraging conformal equivalence and pseudodifferential calculus, advancing the understanding of CR geometry and related problems.

Contribution

It introduces a method to solve the Kohn Laplacian on non-compact CR manifolds using conformal equivalence and extends techniques applicable without boundary assumptions.

Findings

Solved $oxb$ on non-compact CR manifolds via conformal methods.

Established conditions under which $ar{d}_b$ has closed range in $L^2$.

Provided a key step for a positive mass theorem in CR geometry.

Abstract

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve $\square_b$ works whenever $\overline{\partial}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng, Malchiodi and Yang, which they then applied to study the CR Yamabe problem in 3 dimensions.