A Regularity Result for CR Mappings Between Infinite Type Hypersurfaces
Francine Meylan
2008
Communications in Partial Differential Equations
The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M be two real analytic curves in and f a holomorphic function defined on one side of M extending continuously through M and mapping M into M Then f has a holomorphic extension across M In this paper, we extend this classical theorem to higher complex dimensions for a class of hypersurfaces of infinite type. " which should be cited to refer to this work. Definition 0.1. Let M be of infinite type at 0
more »
... n by (0.1) and (0.2). We say that M is m-nondegenerate at 0 if . We shall see (Proposition 1.2) that the above definition is independent of the choice of normal coordinates. It should be noted here that normal coordinates are not unique. The main result of this paper is the following. Theorem 0.2. Let M and M be (germ of) real analytic hypersurfaces of infinite type at 0 in n given by (0.1) and (0.2). Suppose that M (respectively M ) is m-nondegenerate at 0 (respectively m -nondegenerate at 0). Let D ⊂ n be a domain with M in its boundary. Let f M → M be a continuous mapping which is the restriction of a certain continuous mapping over D holomorphic in D with f 0 = 0 Assume that f as a map from M into M is finite to one. Then f extends holomorphically to a neighborhood of 0 Observe that in the case where n = 2 M is m-nondegenerate at 0 if and only if M is of infinite type at 0 and Levi non-flat. The following example shows that Theorem 0.2 fails if M and M are Levi flat. Example 0.3. Let M = z w ∈ 2 Im w = 0 M = z w ∈ 2 Im w = 0 Consider f z w = z + h z w w where h is a holomorphic function defined in the upper half plane in smooth up to the boundary, but which does not extend holomorphically across 0 with h 0 = h 0 = 0 Then one can check that f is a local diffeomorphism near 0 which maps M into M Obviously f does not extend holomorphically at 0 The following example given in [15] shows that Theorem 0.2 may fail if we do not assume that f is finite to one as a map from M into M Example 0.4 [15] . Let M k = z w ∈ 2 Im w = Re w 2 w 2k 2 +k z 2k M k = z w ∈ 2 Im w = Re w 2 z 2k Consider f k z w = zw k+ 1 2 w 2
doi:10.1080/03605300802224680
fatcat:pj5xcdc3uvhffhnmcpnimu4miy