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- Title
Asymptotic expansions and conformal covariance of the mass of conformal differential operators.
- Authors
Ludewig, Matthias
- 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.
- Subjects
DIFFERENTIAL equations; ASYMPTOTIC theory of algebraic ideals; CONFORMAL geometry; ANALYSIS of covariance; DIFFERENTIAL operators; KERNEL (Mathematics)
- Publication
Annals of Global Analysis & Geometry, 2017, Vol 52, Issue 3, p237
- ISSN
0232-704X
- Publication type
Article
- DOI
10.1007/s10455-017-9556-2