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 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.

Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing

“…The zeta-regularized tadpole and the point splitting tadpole appear in the study of conformally covariant elliptic operators such as Yamabe operator and Paneitz operator on a closed Riemanninan manifold Σ [1]. In this context, these functions are known as the mass function of the operators and they are used in the study of mass theorems, regularized traces, and conformal variation of regularized traces, we refer to [40,41,1,34,35,29] for details.…”

The trace and the mass of subcritical GJMS operators

“…7.1]. These operators Q g are formed, following a complicated recipe, out of the derivatives of heat kernel coefficients of the operator L g and hence theoretically can be explicitly computed from the formula given in the proof of Lemma 7.6 in [Lud17]. In practice, however, as n − 2m increases, one needs more and more knowledge of the heat kernel coefficients of the operators in question, which are generally very hard to compute.…”

“…Proof. To explicitly calculate Q g brute force by the formula in the proof of Lemma 7.6 in [Lud17], one needs the knowledge of the heat kernel coefficients Φ 0 and Φ 1 on the diagonal, along with the derivatives of Φ 0 . This is not too involved.…”

“…to obtain a non-infinitesimal version of (2.6). However, in these cases, we could not obtain an explicit formula such as (2.3) yet; already for m = 2, the formula for Q g given in [Lud17] becomes incredibly complicated and involves the second heat kernel coefficient, second derivatives of the first heat kernel coefficient and fourth derivatives of the index zero heat kernel coefficient.…”

“…The proof of Lemma 7.6 of[Lud17] is erroneous in the printed version. For the correct formulas, appeal to the arxiv version.…”