Titel
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 self-adjoint 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.
Stichwort
Filtered manifoldHypoelliptic operatorHeat kernel expansionZeta functionNon-commutative residueGeneric rank-two distribution in dimension
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
https://phaidra.univie.ac.at/o:1078856
Erschienen in
Titel
The Journal of Geometric Analysis
Band
30
Ausgabe
1
Seitenanfang
337
Seitenende
389
Verlag
Springer Science and Business Media LLC
Projektnummer
Y963-N35; P30233 – Austrian Science Fund (FWF)
Erscheinungsdatum
2019
Zugänglichkeit
Rechteangabe
© The Author(s) 2019

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