The Reflection Map and Infinitesimal Deformations of Sphere Mappings
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The Reflection Map and Infinitesimal Deformations of Sphere Mappings Michael Reiter1 Received: 14 June 2019 © The Author(s) 2019
Abstract The reflection map introduced by D’Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided. Keywords CR geometry · CR mapping · Infinitesimal deformations · Reflection map · Unit sphere Mathematics Subject Classification 32V40 · 32V30
1 Introduction The main motivation is the study of real-analytic CR maps of the unit sphere S 2n−1 in Cn for n ≥ 2, which is defined by S 2n−1 = {z = (z 1 , . . . , z n ) ∈ Cn : z2 = |z 1 |2 + . . . + |z n |2 = 1}. For n = 2 write z = z 1 , w = z 2 . A lot is known about mappings of spheres, see the survey by D’Angelo [9] and the references therein. A prominent example of a sphere map is the homogeneous sphere map Hnd of degree d from S 2n−1 into S 2K −1 for some K = K (n, d) ∈ N, which consists of all lexicographically ordered monomials in z = (z 1 , . . . , z n ) ∈ Cn of degree d and is given by
The author was supported by the Austrian Science Fund (FWF), project P28873-N35.
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Michael Reiter [email protected] Faculty of Mathematics, University of Vienna, Vienna, Austria
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M. Reiter
Hnd (z)
α α z = d
. |α|=d
The purpose of this article is to study the reflection map, which was introduced by D’Angelo [7] in the case of sphere mappings and further investigated by the same author in [8] in the case of maps of hyperquadrics. The reflection map of a mapping H allows to effectively compute and deduce several properties of the X-variety associated to H . The X-variety was introduced and studied by Forstneriˇc in [19] to extend CR maps satisfying certain smoothness assumptions. In the case of real-analytic CR maps of spheres it is shown that these maps are rational. The homogeneous sphere map Hnd plays a crucial role in the classification of polynomial maps, see the works of D’Angelo [3,5] and [10] for rational sphere maps. The homogeneous sphere map appears in the definition of the reflection map C H for a rational sphere map H = P/Q : S 2n−1 → S 2m−1 with Q = 0 on S 2n−1 : Let VH : Cm → C K be a matrix with holomorphic entries, satisfying VH (X ) · H¯ nd / Q¯ = X · H¯ on S 2n−1 for X ∈ Cm , where · denotes the euclidean inner product. The previous identity is achieved by the homogenization technique of D’Angelo [3]. VH is referred to as reflection matrix and C H (X ):=VH (X ) · H¯ nd / Q¯ for X ∈ Cm . See Sect. 2.4 below for more details. In this article the reflection matrix will be applied in two ways. In the first case it is shown that nondegeneracy conditions of sphere maps can be r
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