Spectral invariants of CR manifolds
type: Item
instance of: scholarly article
title: Spectral invariants of CR manifolds
zbMATH Open document ID: 0685.58033
author: Nancy K. Stanton
published in: Michigan Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let P be a self-adjoint elliptic second order differential operator on a smooth manifold M without boundary. Then the trace of the heat kernel has an asymptotic expansion as \(t\to 0^+\) of the form \[ (1)\quad Tr_{L^ 2}(e^{-tP})\sim \sum_{n}a_ n(P)t^{(n-m)/2}. \] Let \(\tau\) be the scalar curvature and let \(P=d\delta +(m-2)/4(m-1)\tau\) be the conformal Laplacian. \textit{T. Parker} and \textit{S. Rosenberg} [J. Differ. Geom. 25, 199-222 (1987; Zbl 0644.53038)] and \textit{T. P. Branson} and \textit{B. Oersted} [Compos. Math. 60, 261-203 (1986; Zbl 0608.58039)] show the coefficient of \(t^ 0\), \(a_ m(P)\), is a global conformal invariant and the coefficient of \(t^{-1}\), \(a_{m-2}(P)\), is the integral of a local conformal invariant. Let M be a strictly pseudoconvex CR manifold equipped with a Levi metric. If P is the sublaplacian \(box_ b\) acting on forms of a suitable degree, there is a suitable analogue of (1). If P is the pseudoconformal Laplacian - the CR analogue of the conformal Laplacian - the author shows the coefficient of \(t^ 0\) is a conformal invariant and the coefficient of \(t^{-1}\) is the integrand of a local CR invariant. This constructs interesting new CR invariants. These invariants vanish on spheres and in dimension 3.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4122899
zbMATH Keywords: self-adjoint elliptic second order differential operator
zbMATH Keywords: heat kernel
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: conformal Laplacian
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003949
OpenAlex ID: W2065182482
DOI: 10.1307/MMJ/1029003949
description: scientific article; zbMATH DE number 4122899