A Bishop surface with a vanishing Bishop invariant

Abstract

We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant s < ∞ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real analytic Bishop surface near a complex tangent with a vanishing Bishop invariant can not be determined by a finite part of the Taylor expansion of its defining equation. This answers, in the negative, a problem raised by J. Moser in 1985 after his joint work with Webster in 1983 and his own work in 1985. Such a phenomenon is strikingly different from the celebrated theory of Moser-Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants. We also show that a formal map between two real analytic Bishop surfaces with the Bishop invariant λ = 0 and with the Moser invariant s = ∞ is convergent. Hence, two real analytic Bishop surfaces with λ = 0 and s < ∞ are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Notice that there are many non-convergent formal transformations between Bishop surfaces with λ = 0 and s = ∞. Notice also that a generic formal map between two real analytic hyperbolic Bishop surfaces is divergent as shown by Moser-Webster and Gong. Hence, Bishop surfaces with a vanishing Bishop invariant and s = ∞ behave very differently, in this respect, from hyperbolic Bishop surfaces or elliptic Bishop surfaces with λ = 0 and s = ∞. We also show that a Bishop surface with λ = 0 and s < ∞ generically has a trivial automorphism group and has the largest possible automorphism group if and only if it is biholomorphic to the model surface M s = {(z, w) ∈ C 2 : w = |z| 2 +z s +z s }. Notice that, by the Moser-Webster theorem, an elliptic Bishop surface with λ = 0, always has automorphic group Z 2 . Hence, Bishop surfaces with λ = 0 and s = ∞ have the similar character as that of strongly pseudoconvex real hypersurfaces in the complex spaces of higher dimensions. * Supported in part by NSF-0500626

Equivalence problem for Bishop surfaces

“…The second problem concerns the higher order invariants for Bishop surfaces with λ = 0. It was answered in a recent paper of the authors [24]. The way we constructed the normal form is to directly work with the mappings from Bih 0 (C 2 , 0).…”

Normal forms for CR singular codimension-two Levi-flat submanifolds

“…Bishop's work on the family of attached analytic discs has been refined by Kenig-Webster [25,26], Huang-Krantz [21], and Huang [20]. The normal form theory for real submanifolds for Bishop surfaces or submanifolds was established by MoserWebster [31]; see also Moser [30], Gong [17][18][19], Huang-Yin [22], and Coffman [10]. We would like to mention that the Moser-Webster normal form does not deal with the case of vanishing Bishop invariant.…”

“…The formal normal form and its application to holomorphic classification for surfaces with vanishing Bishop invariant was achieved by Huang-Yin [22] by a completely different method. Real submanifolds with complex tangents have been studied in other situations.…”

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

“…Since Poincaré's celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4,6,7,11,13,14,22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence.…”