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

TL;DR

This paper provides an explicit asymptotic expansion of the Schwartz kernel for complex powers of conformally invariant differential operators on compact manifolds, linking the constant term to the local zeta function and exploring conformal invariance of the mass.

Contribution

It offers a detailed description of the asymptotic expansion of complex powers of conformally invariant operators and characterizes the conformal invariance of the associated mass in various cases.

Findings

The constant term in the expansion equals the local zeta function of L.

The mass is conformally invariant when the dimension is odd and kernel is zero.

The paper describes how conformal invariance fails otherwise.

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.