The heat asymptotics on filtered manifolds

TL;DR

This paper develops a universal heat kernel expansion for hypoelliptic operators on filtered manifolds, extending classical results to more general geometric structures and providing tools for spectral analysis.

Contribution

It establishes a universal heat kernel expansion for Rockland operators on filtered manifolds and introduces a generalized calculus with a non-commutative residue.

Findings

Derived heat trace asymptotics for hypoelliptic operators

Extended Weyl's law to filtered manifold settings

Provided a McKean-Singer type index formula

Abstract

The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universal heat kernel expansion for formally selfadjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion implies that the new calculus, a more general version of the Heisenberg calculus, also has a non-commutative residue. Many of the well known implications of the heat expansion such as, the structure of the complex powers, the heat trace asymptotics, the continuation of the zeta function, as well as Weyl's law for the eigenvalue asymptotics, can be adapted to this calculus. Other consequences include a McKean-Singer type formula for the index of Rockland differential operators. We illustrate some of these results by providing a more explicit description of Weyl's law for Rumin-Seshadri operators associated with curved BGG sequences over 5-manifolds equipped with a rank two distribution of Cartan type.