Strictly pseudoconvex domains in \({\mathbb{C}}^ n\). (English) Zbl 0546.32008
Bull. Am. Math. Soc., new Ser. 8, 125-322 (1983) and The mathematical heritage of Henri Poincaré, Proc. Symp. Pure Math. 39, Part 1, Bloomington/Indiana 1980, 189-386 (1983).
[This article was published twice, the second time in the book Zbl 0512.00008.]
These long lecture notes on geometry and analysis on strictly pseudoconvex domains in \({\mathbb{C}}^ n\) cover essentially the part of the theory which is closely related with PDE and differential geometry. The first chapters give an exposition of fundamental facts about elliptic partial differential equations, pseudodifferential operators, Fourier integral operators, elementary differential geometry and the heat equation. The following chapters are devoted to strictly pseudoconvex domains with real-analytic boundaries. Chapter 7 begins with a study of the \({\bar\partial }_ b\)-equation on the Heisenberg group and of the \({\bar\partial }\)-Neumann problem on the Siegel domain \(\{Im z_{n+1}>| z_ 1|^ 2+...+| z_ n|^ 2\}\subset {\mathbb{C}}^{n+1};\) the results of Folland and Stein for general strictly pseudoconvex domains are sketched. Chapter 8 (”Ellipsoids and the method of reflections”) presents Fefferman’s result on continuation up to the boundary of biholomorphic mappings of strictly pseudoconvex domains and theorems of Webster for domains with real algebraic boundaries. Chapters 9 and 10 are devoted to Moser’s normal form and Chern-Moser invariants. The last two chapters are devoted to the complex Monge-Ampère equation, Fefferman’s chains and light rays over the boundary, and the asymptotic expansion of the Bergman kernel.
The main theme of these notes is the analogy between strongly pseudoconvex domains and Riemannian manifolds, Bergman kernel and heat kernel,... This is perhaps the reason why other tools of analysis in strongly pseudoconvex domains, especially integral formulas for \({\bar\partial }\) and \({\bar\partial }_ b\), are never mentioned in these notes, even in the bibliography. Recent work of I. Lieb and R. M. Range [Math. Ann. 225, 221-251 (1983; Zbl 0504.32015); Math. Ann. 266, 449-456 (1984; Zbl 0513.32008); Bull. Am. Math. Soc. 11, 355- 358 (1984)] has shown that these tools can also be applied to the \({\bar\partial }\)-Neumann and related problems.
These long lecture notes on geometry and analysis on strictly pseudoconvex domains in \({\mathbb{C}}^ n\) cover essentially the part of the theory which is closely related with PDE and differential geometry. The first chapters give an exposition of fundamental facts about elliptic partial differential equations, pseudodifferential operators, Fourier integral operators, elementary differential geometry and the heat equation. The following chapters are devoted to strictly pseudoconvex domains with real-analytic boundaries. Chapter 7 begins with a study of the \({\bar\partial }_ b\)-equation on the Heisenberg group and of the \({\bar\partial }\)-Neumann problem on the Siegel domain \(\{Im z_{n+1}>| z_ 1|^ 2+...+| z_ n|^ 2\}\subset {\mathbb{C}}^{n+1};\) the results of Folland and Stein for general strictly pseudoconvex domains are sketched. Chapter 8 (”Ellipsoids and the method of reflections”) presents Fefferman’s result on continuation up to the boundary of biholomorphic mappings of strictly pseudoconvex domains and theorems of Webster for domains with real algebraic boundaries. Chapters 9 and 10 are devoted to Moser’s normal form and Chern-Moser invariants. The last two chapters are devoted to the complex Monge-Ampère equation, Fefferman’s chains and light rays over the boundary, and the asymptotic expansion of the Bergman kernel.
The main theme of these notes is the analogy between strongly pseudoconvex domains and Riemannian manifolds, Bergman kernel and heat kernel,... This is perhaps the reason why other tools of analysis in strongly pseudoconvex domains, especially integral formulas for \({\bar\partial }\) and \({\bar\partial }_ b\), are never mentioned in these notes, even in the bibliography. Recent work of I. Lieb and R. M. Range [Math. Ann. 225, 221-251 (1983; Zbl 0504.32015); Math. Ann. 266, 449-456 (1984; Zbl 0513.32008); Bull. Am. Math. Soc. 11, 355- 358 (1984)] has shown that these tools can also be applied to the \({\bar\partial }\)-Neumann and related problems.
Reviewer: G.Roos
MSC:
| 32T99 | Pseudoconvex domains |
| 32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |
| 35K05 | Heat equation |
| 35N15 | \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs |
| 32V40 | Real submanifolds in complex manifolds |
| 32D15 | Continuation of analytic objects in several complex variables |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
Keywords:
d-bar-problem; strictly pseudoconvex domains; elliptic partial differential equations; pseudodifferential operators; integral operators; heat equation; reflections; continuation; Chern-Moser invariants; Bergman kernelReferences:
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