The Heat Asymptotics on Filtered Manifolds

Area of Science:

• Differential Geometry
• Analysis on Manifolds
• Mathematical Physics

Background:

• The heat kernel expansion connects local and global geometric features of manifolds.
• Rockland operators, common in geometric structures, are hypoelliptic.
• Existing methods are limited for certain geometric structures and operators.

Purpose of the Study:

• To establish a universal heat kernel expansion for self-adjoint, non-negative Rockland operators on closed filtered manifolds.
• To adapt implications of heat expansion, such as complex powers and spectral asymptotics, to a new calculus.
• To derive a McKean-Singer type formula for the index of Rockland operators.

Main Methods:

• Analysis of parametrices within a newly developed calculus tailored for Rockland operators.
• Construction of a generalized Heisenberg calculus adapted to filtered manifolds.
• Application of heat kernel expansion techniques to Rockland operators.

Main Results:

• A universal short-time heat kernel expansion for Rockland operators on filtered manifolds.
• Demonstration that the new calculus possesses a non-commutative residue.
• Adaptation of key implications of heat expansion, including spectral asymptotics (Weyl's law) and zeta function continuation.

Conclusions:

• The developed calculus provides a framework for studying Rockland operators analogous to classical heat kernel methods.
• The results extend spectral theory and index theory to a broader class of geometric structures.
• Explicit formulas for Weyl's law are derived for specific operators like Rumin-Seshadri operators.