The heat asymptotics on filtered manifolds

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.

v2: accepted manuscript, introduction rewritten, results unchanged, added journal reference and DOI

References in corpus (7)

• Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition
• On the tangent groupoid of a filtered manifold
• The twistor spinors of generic 2- and 3-distributions
• On Nurowski's conformal structure associated to a generic rank two distribution in dimension five
• Tangent Maps and Tangent Groupoid for Carnot Manifolds
• The Geometry of Almost Einstein (2,3,5) Distributions
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Cited by in corpus (5)

• Graded hypoellipticity of BGG sequences
• Analytic torsion of generic rank two distributions in dimension five
• Analytic torsion of nilmanifolds with (2,3,5) distributions
• Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group
• Shubin calculi for actions of graded Lie groups