On the Cauchy problem in complex analysis
The authors are interested in developing a global theory of the cohomology of a \(q\)-pseudoconcave \(CR\) manifold \(M.\) In the approach to the subject a central role is played by the Cauchy problem. This amounts to a study of the map, induced by restriction, from the cohomology of a complex neighborhood \(U\) of \(M\) to the cohomology of \(M.\) Surjectivity and injectivity of this map correspond precisely to existence and uniqueness for this Cauchy problem. The surjectivity of this map was studied in \textit{C. D. Hill and M. Nacinovich} [Math. Z. 220, No. 3, 347-367 (1995; Zbl 0843.32011)] by the technique of aneurysms. This led, on the one hand, to theorems about the finite dimensionality or vanishing for these cohomology groups, and on the other hand, to comparison theorems relating the different kinds of cohomology groups that can be attached to \(M.\) Some of these groups are a priori different because of the nonvalidity of the Poincaré lemma in certain intermediate ranges. In this paper, by studying the injectivity of the same maps, the authors manage to establish the equivalence between the smooth and the sheaf cohomology groups of \(M\) except in the intermediate range where they may differ. These results also yield information about existence in a related Cauchy problem having zero initial data, as well as a global uniqueness result which is reminiscent of the Holmgren uniqueness theorem.
scientific article; zbMATH DE number 1000247
scientific article