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The heat equation for the ${\bar \partial}$-Neumann problem. II. (English) Zbl 0715.35056
[For part I see Commun. Partial Differ. Equations 12, 351-413 (1987; Zbl 0615.58033).] Let ${\bar \Omega}$ be a compact complex $(n+1)$-dimensional Hermitian manifold with smooth boundary M and let $\square\sb{p,q}$ denote the ${\bar \partial}$-Laplacian on (p,q) forms on ${\bar \Omega}$, which satisfy the ${\bar \partial}$-Neumann boundary conditions. In part I the asymptotic expansion of tr exp(-t$\square\sb{p,q})$ as $t\to +0$ was obtained. In this part II the authors go on to show that the coefficients in the trace expansion are integrals of local geometric invariants. Suppose that at each point of M, ${\bar \Omega}$ is either strictly pseudoconvex or strictly pseudoconcave. Suppose also that ${\bar \Omega}$ is equipped with a Hermitian metric which is Kähler in a neighborhood of the boundary M and which induces a Levi metric on M. Then for $1\le q\le n-1$ $$ tr \exp (-t\square\sb{p,q})\sim t\sp{-n- 1}\{\sum\sp{\infty}\sb{j=0}b\sb jt\sp j+c\sb 0+\sum\sp{\infty}\sb{j=1}(c\sb j+c'\sb j \log t)t\sp{(1/2)j}\}\quad as\quad t\to +0. $$ If ${\bar \Omega}$ is strictly pseudoconvex (pseudoconcave) then the expansion also holds for $q=n$ $(q=0)$. The coefficients $b\sb j$ are integrals over $\Omega$ of universal polynomials in the components of the curvature and torsion of the Hermitian connection and their covariant derivatives with respect to this connection. The coefficients $c\sb j$ and $c'\sb j$ are integrals over M of universal polynomials in the components of the second fundamental form of M, the curvature and torsion of the Webster-C. M. Stanton connection, and their covariant derivatives with respect to this connection, as well as the components of the Hermitian curvature of ${\bar \Omega}$ and its Hermitian covariant derivatives on M.
Reviewer: N.N.Tarkhanov

35N15$\bar\partial$-Neumann problem and generalizations; formal complexes
58J35Heat and other parabolic equation methods for PDEs on manifolds
35K05Heat equation
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