Characteristic Laplacian in sub-Riemannian geometry

TL;DR

This paper investigates a Laplacian operator in sub-Riemannian geometry, revealing its limitations such as lack of hypoellipticity and incompatibility with form bigrading, impacting related conjectures.

Contribution

It introduces and analyzes a characteristic Laplacian in sub-Riemannian geometry, highlighting its non-hypoelliptic nature and implications for characteristic cohomology.

Findings

The Laplacian is not hypoelliptic in general.

It does not preserve the bigrading on forms.

Implications for Griffiths' conjecture are discussed.

Abstract

We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting. We also discuss the consequences of these negative results for a conjecture of P. Griffiths, concerning the characteristic cohomology of period domains.