Hypersurface In Cn Research Articles

A kernel approach to the local solvability of the tangential Cauchy Riemann equations

An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the Y ( q ) Y(q) condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in C n {{\mathbf {C}}^n} . In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when Y ( q ) Y(q) is not satisfied.

ESTIMATES FOR THE RADIUS OF CONVERGENCE OF POWER SERIES DEFINING MAPPINGS OF ANALYTIC HYPERSURFACES

In this article the authors obtain lower bounds for the radius of convergence of power series which define a mapping from one nondegenerate real analytic hypersurface in Cn to another. For certain classes of surfaces a complete list is given of the parameters which substantially influence the size of the radius of convergence. In particular, for compact hypersurfaces with positive definite Levi form the radius is bounded by a constant depending on the pair of surfaces and not on the mapping. Bibliography: 5 titles.

Projective connections in CR geometry

Holomorphic invariants of an analytic real hypersurface in ℂn+1 can be computed by several methods, coefficients of the Moser normal form [4], pseudo-con-formal curvature and its covariant derivatives [4], and projective curvature and its covariant derivatives [3]. The relation between these constructions is given in terms of reduction of the complex projective structure to a real form and exponentiation of complex vectorfields to give complex coordinate systems and corresponding Moser normal forms. Although the results hold for hypersurfaces with non-degenerate Levi-form, explicit formulas will be given only for the positive definite case.

Affine connections and defining functions of real hypersurfaces in 𝐶ⁿ

The affine connection and curvature introduced by Tanaka on a strongly pseudoconvex real hypersurface are computed explicitly in terms of its defining function. If Fefferman’s defining function is used, then the Ricci form is shown to be a function multiple of the Levi form. The factor is computable by Fefferman’s algorithm and its positivity implies the vanishing of certain cohomology groups (of the ∂ ¯ b {\bar \partial _b} complex) in the compact case.

Some boundary properties of holomorphic functions of several complex variables

A local uniqueness theorem and analogs of the theorem on removable singularities under the hypothesis of boundedness are proved for functions satisfying the tangential Cauchy-Riemann conditions on hypersurfaces in Cn. The results can be interpreted as giving certain boundary properties of holomorphic functions of several complex variables.