The index of sub-laplacians: beyond contact manifolds

TL;DR

This paper investigates the index theory of sub-Laplacian operators on Carnot manifolds with higher nilpotency, revealing that in many cases, the index theory is trivial, contrasting with the richer theory in lower degrees.

Contribution

It extends the understanding of sub-Laplacian index theory to higher nilpotency degrees, providing numerous examples where the index is trivial, thus broadening the scope beyond contact and polycontact manifolds.

Findings

Index theory is trivial in many higher nilpotency cases.

Rich index theory exists for contact and polycontact manifolds.

Examples demonstrate the limitations of index theory in higher degrees.

Abstract

In this paper we study the following question: do sub-Laplacian type operators have non-trivial index theory on Carnot manifolds in higher degree of nilpotency? The problem relates to characterizing the structure of the space of hypoelliptic sub-Laplacian type operators, and results going back to Rothschild-Stein and Helffer-Nourrigat. In two degrees of nilpotency, there is a rich index theory by work of van Erp-Baum on contact manifolds, that was later extended to polycontact manifolds by Goffeng-Kuzmin. We provide a plethora of examples in higher degree of nilpotency where the index theory is trivial.