Jerison-Lee identity and Semi-linear subelliptic equation on CR manifold

TL;DR

This paper develops a theoretical framework using invariant tensors and dimensional conservation to derive new differential identities for subelliptic equations on CR manifolds, leading to rigidity results and sharp inequalities.

Contribution

It introduces a novel theoretical approach to identify differential identities for subelliptic equations on CR manifolds, extending previous computational findings.

Findings

New differential identities for subelliptic equations on CR manifolds

Rigidity results showing uniqueness of solutions under certain conditions

Derivation of the sharp Folland-Stein inequality on closed CR manifolds

Abstract

In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR) Yamabe problem, Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg group $\mathbb H^n$ by using the computer in [11]. They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae. With the help of dimensional conservation and invariant tensors, we can answer the above question. For a class of subcritical exponent subelliptic equations on the CR manifold, several new types of differential identities are found. Then we use those identities to get the rigidity result, where rigidity means that subelliptic equations have no other solution than some constant at least when parameters are in a certain range. The rigidity result also deduces the sharp Folland-Stein inequality on closed CR manifolds.