Henkin, Gennadi Markovič; Leiterer, Jürgen Andreotti-Grauert theory by integral formulas. (English) Zbl 0654.32001 Mathematical Research 43. Berlin (GDR): Akademie-Verlag. 270 p. (1988). The Andreotti-Grauert theory is the theory of the cohomology of q-convex and q-concave manifolds. This monograph develops this theory from the viewpoint of explicit integral representation for solution of the Cauchy- Riemann equation for differential forms on strictly q-convex and q- concave domains. This method give us new proofs of remarkable results with uniform estimates. This results are: uniform approximation and uniform interpolation for \({\bar \partial}\)-cohomology classes on strictly q-convex domains, solution of the Levi problem for \({\bar \partial}\)-cohomology with uniform estimate, Andreotti-Vesentini separation theorem of the Dolbeault cohomology of order q on q-concave manifolds, the Rossi theorem on attaching complex manifolds along a strictly pseudoconcave boundary of real dimension \(\geq 5.\) Brief contents. Chapter 1. Integral formulas and first applications. Chapter 2. q-convex and q-concave manifolds. Chapter 3. The Cauchy- Riemann equations on q-convex manifolds. Chapter 4. The Cauchy-Riemann equations on q-concave manifolds. Chapter 5. Some applications. Reviewer: S.Krendelev Cited in 53 Documents MSC: 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32F10 \(q\)-convexity, \(q\)-concavity 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators Keywords:Andreotti-Grauert theory; cohomology of q-convex and q-concave manifolds; explicit integral representation; uniform estimates; uniform approximation; uniform interpolation; Levi problem; Andreotti-Vesentini separation theorem; Rossi theorem; Cauchy-Riemann equations PDFBibTeX XMLCite \textit{G. M. Henkin} and \textit{J. Leiterer}, Andreotti-Grauert theory by integral formulas. Berlin (GDR): Akademie-Verlag (1988; Zbl 0654.32001)