Spectral invariants of CR manifolds
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.
self-adjoint elliptic second order differential operator
scientific article; zbMATH DE number 4122899