Regularity of mappings into classical domains
- PDF / 595,340 Bytes
- 39 Pages / 439.37 x 666.142 pts Page_size
- 85 Downloads / 169 Views
Mathematische Annalen
Regularity of mappings into classical domains Ming Xiao1 Received: 7 May 2018 / Revised: 20 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study the regularity and algebraicity for mappings into classical domains. Among other things, we establish everywhere regularity and algebraicity results for C 2 CR maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large. Mathematics Subject Classification Primary 32H40; Secondary 32M15 · 32H35
1 Introduction The first part of the article is devoted to establishing reflection principle type results for holomorphic and CR maps in several complex variables. The typical question of the reflection principle asks to find conditions under which a CR map between real analytic CR submanifolds in complex spaces extends holomorphically to an open neighborhood of the source manifold and when the submanifolds are merely smooth, we investigate when the map has C ∞ regularity. Results of this type date back to the work of Fefferman [18], Lewy [41], and Pinchuk [54]. In this paper, we concentrate on CR mappings between real hypersurfaces of different dimensions. Much attention has been paid to the development of the reflection principle along this line since the pioneering work of Webster [65], Faran [17], and Forstneriˇc [20]. We cannot mention all related work but only name a few recent work here: [5,6,35,38–40,43]. See the book by Baouendi–Ebenfelt–Rothschild [3] for a detailed account and more earlier references on this subject. A particular case of interest is to study reflection principle type problem for maps between real algebraic CR manifolds. A CR manifold is called real algebraic if it is defined by real polynomials. The class of real algebraic CR manifolds is of
Communicated by Ngaiming Mok.
B 1
Ming Xiao [email protected] Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA
123
M. Xiao
fundamental importance in several complex analysis. They arises naturally as the boundaries of bounded symmetric domains and the tube domain of future light cone, as well as some homogeneous CR manifolds. Algebraicity properties of biholomorphisms between real algebraic CR submanifolds has been extensively studied (cf. [2,3] and references therein). Starting from the work of Forstneriˇc [20] and Huang [26], one expects a finitely smooth CR map between real algebraic CR manifolds of different dimensions also to be algebraic under some geometric conditions. Here a holomorphic map F : U ⊂ Cn → Cm is called (Nash) algebraic if each component of F satisfies some holomorphic polynomial equation. Huang [26] proved a CR map F : M ⊂ Cn → M ⊂ C N (N > n > 1) of class C N −n+1 between strongly pseudoconvex real algebraic hypersurfaces M and M must extend to an algebraic holomorphic map. In particular, if M and M are spheres, then by the result of Forstneriˇc [20], F must extend to a holomorphic rational map. Zaitsev [69] established algebraicity resu
Data Loading...
