Asymptotic Expansions and Conformal Covariance of the Mass of Conformal Differential Operators

Abstract

We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of $m$-Laplace type operators $L$ on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns turns out to be given by the local zeta function of $L$. In particular, the constant term in the asymptotic expansion of the Green's function $L^{-1}$ is often called the mass of $L$, which (in case that $L$ is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant $m$-Laplace operators $L$ (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of $M$ is odd and that $\ker L=0$, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.

38 pages, fixed a mistake in proof of Lemma 7.6

References in corpus (3)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• Explicit formulas for GJMS-operators and $Q$-curvatures
• Heat kernel expansions, ambient metrics and conformal invariants

Cited by in corpus (2)

• Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing
• The Trace and the Mass of subcritical GJMS Operators