The heat equation on a CR manifold
The trace of the heat semigroup for the Laplacian on a compact oriented Riemannian manifold has an asymptotic expansion in powers of the time t, for small positive t, whose coefficients are integrals of local geometric invariants. In this paper, the authors prove analogous results for the sublaplacian \(\square_ b\) on (p,q) forms on a compact oriented Hermitian CR manifold satisfying condition Y(q). To prove their results, they use pseudodifferential calculus. Because \(\square_ b\) is not elliptic, the classical pseudodifferential calculus does not apply, so they develop an appropriate pseudodifferential calculus for the problem. The main result is the following. Theorem. Let M be a compact oriented Hermitian CR manifold of dimension \(2n+1\) and suppose the Levi form satisfies condition Y(q) at each point of M. let \(\square_{b,q}\) denote the \(\partial_ b\)-Laplacian on (p,q) forms on M. Then \[ tr(e^{-t\square_{b,q}})\sim t^{- (n+1)}\sum^{\infty}_{j=0}k_ jt^ j\quad as\quad t\to 0^+ \] where \(k_ j=\int_{M}K_ j(x)dV(x)\); \(K_ j(x)\) is a locally computable function on M. In the special case that M is strictly pseudoconvex, so Y(q) is satisfied for \(0<q<n\), and that the metric is a Levi metric, the function \(K_ j\) may be calculated at \(x\in M\) by evaluating a universal polynomial (depending only on n, p, q and j) in the components of the curvature and torsion of the Webster-C. Stanton connection and their covariant derivatives, computed in normal coordinates at x.
scientific article; zbMATH DE number 3881538
scientific article