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.
Highlights
Let (M, g) be a compact Riemannian manifold of dimension n and let L be a self-adjoint m-Laplace type operator, acting on sections of a metric vector bundle V over M. By this we mean that L is a differential operator of order 2m such that the principal symbol of L equals the principal symbol of (∇∗∇)m for some connection on M
Examples of higher-order operators which are of this type are the GJMS operators, which play a prominent role in conformal geometry
The difference of the two sides is a continuous function near the diagonal and can be evaluated there. We refer to this value as the constant term in the asymptotic expansion of L−s(x, y)
Summary
Let (M, g) be a compact Riemannian manifold of dimension n and let L be a self-adjoint m-Laplace type operator, acting on sections of a metric vector bundle V over M. The difference of the two sides is a continuous function near the diagonal and can be evaluated there We refer to this value as the constant term in the asymptotic expansion of L−s(x, y). Theorem 6.6 completes this picture, by stating that in odd dimensions (where ζg(s, x) has no pole at s = 1), the value of the zeta function itself is conformally covariant, with the same transformation law as the residue in even dimensions. 4, we introduce the relevant concepts of global analysis, to set notation: general m-Laplace type operators L on compact Riemannian manifolds, as well as their complex powers, the Minakshisundaram-Pleijel asymptotic expansion of their heat kernel and the corresponding local zeta function. We turn to conformally covariant m-Laplace type operators and give a further discussion of Theorem 6.6, the positive mass conjecture and related questions.
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