A Bishop surface with a vanishing Bishop invariant

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 6= ∞ 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 6= ∞ 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 Ms = {(z, w) ∈ C : w = |z|2+zs+zs}. Notice that, by the Moser-Webster theorem, an elliptic Bishop surface with λ 6= 0, always has automorphic group Z2. Hence, Bishop surfaces with λ = 0 and s 6= ∞ have the similar character as that of strongly pseudoconvex real hypersurfaces in the complex spaces of higher dimensions. ∗Supported in part by NSF-0500626