2016/12/31 by Matthias Ludewig · 3 citations
Mathematics · #Asymptotic expansion #Conformal geometry #Conformal map #Conformal symmetry #Differential geometry #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Invariant (physics) #Laplace operator #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Riemannian manifold #Spectral Theory in Mathematical Physics #Weyl transformation #math.AP #math.DG
paper · pdf · doi:10.1007/s10455-017-9556-2
published in Annals of Global Analysis and Geometry 52(3), 237-268 (Springer Science+Business Media) · 38 pages, fixed a mistake in proof of Lemma 7.6
openalex publication_date 2017/04/07 · arxiv created 2017/04/26 · arxiv updated 2022/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.