The stationary disc method in the unique jet determination of CR automorphisms
Finite jet determination of holomorphic maps of real manifolds has gained notable attention during the recent decades. Among them, we have in particular the impressive result of \textit{S. S. Chern} and \textit{J. Moser} in their celebrated work [Acta Math. 133, 219--271 (1974; Zbl 0302.32015)] showing that every holomorphic automorphism of a certain real-analytic nondegenerate hypersurface in an arbitrary complex space, which preserves some nondegenerate point \(p\) is uniquely determined by its jets of order two at this point. One finds a large amount of considerable works on the finite jet determination of holomorphic maps in the real analytic setting. In the smooth case, most results on this issue rely on the method of complete differential systems initiated by Cartan and Chern-Moser. But in the case of \textit{finitely smooth} manifolds, it seems that the method of stationary discs is the only known way to treat the problem of finite jet determination. Attached to a given real submanifold \(M\subset\mathbb{C}^N\), the stationary discs are actually a family of analytic invariant objects which admit a lift with a pole of order at most 1 at the origin. The main idea behind attaching analytic stationary discs to real submanifolds is a boundary value problem, namely a nonlinear problem of Riemann-Hilbert type. The paper under review studies the finite jet determination of holomorphic automorphisms of some specific, but interesting, (degenerate and nondegenerate) kinds of real submanifolds in arbitrary complex spaces by constructing the attached stationary discs and analyzing their corresponding geometric features.
scientific article; zbMATH DE number 7346309
scientific article