Hypersurface In C3 Research Articles

Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in ℂ3

Abstract Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.

Higher order symmetries of real hypersurfaces in ℂ³

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By results of Kolar, Meylan, and Zaitsev in 2014, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in C 2 \mathbb C^2 . We analyze this phenomenon for hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author for an AIM workshop in 2010.

The sphere in C2 as a model surface for degenerate hypersurfaces in C3

In the paper, an unexpected correspondence between the automorphisms of 5D real uniformly 2-nondegenerate hypersurfaces of the space C3 and the automorphisms of the 3D hypersphere in C2 is constructed. In a certain sense, the 3D hypersphere, which is, as is known, a model surface for the class of nondegenerate 3D hypersurfaces in C2, has this status also with respect to the above class of 5D hypersurfaces in C3.

О размерностях групп аффинных преобразований, транзитивно действующих на вещественных гиперповерхностях в C3

О размерностях групп аффинных преобразований, транзитивно действующих на вещественных гиперповерхностях в C3

Homogeneous Hypersurfaces in ℂ3, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ3 has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ3 with finite-dimensional infinitesimal automorphism algebra \(\mathfrak{g}\) and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all \(\mathfrak{g}\)-homogeneous hypersurfaces in ℂ3 and also classify the corresponding local transitive actions of the model algebra \(\mathfrak{g}\) on hypersurfaces in ℂ3.

An example of a two-parameter family of affine homogeneous real hypersurfaces in ℂ3

This paper studies real hypersurfaces in multidimensional complex spaces homogeneous with respect to holomorphic transformations. In the case of spaces of two complex variables, this problem was completely solved in explicit form as early as in 1932 by Cartan [1]. But even in the problem of describing holomorphic homogeneous real hypersurfaces in complex 3-spaces, there are many questions that remain unanswered so far. Therefore, it is natural to first study the simpler problem of affine homogeneous hypersurfaces in complex space C3. We emphasize that the family of homogeneous real hypersurfaces in C3 is very large (see [2], [3]). For this reason, an important part of the problems stated above is not only finding new families of homogeneous manifolds but also determining their position in some (desirably, complete) classification. Results of the papers [2] and [3] mentioned above show that both the holomorphic and the affine homogeneity of hypersurfaces in multidimensional spaces can be studied within the framework of a unified approach related to local canonical equations of the manifolds under consideration. For example, in [3], by analogy with the well-known (holomorphic)Moser normal form [4], the notion of an affine canonical equation of a real hypersurface in C3 was suggested. According to [3], in a neighborhood of any point, the equation of a strictly pseudoconvex (SPC) real-analytic (not necessarily homogeneous) hypersurface in this space can be reduced by affine transformations to the form

On Chern–Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces M at the origin in Cn+1 for which the group of local CR-automorphisms preserving the origin has dimension d0(M) equal to either n 2 − 2n + 1 with n ≥ 2, or n2 − 2n with n ≥ 3. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in C3 with d0(M) = 1, 2 due to A. Loboda, and complement earlier joint work by V. Ezhov and the author for the case d0(M) ≥ n 2 − 2n + 2.

A new example of a uniformly Levi degenerate hypersurface in C3

We present a homogeneous real analytic hypersurface in C3, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube ΓC over the two-dimensional real cone in R3 is also homogeneous and has a seven-dimensional CR automorphism group. However, our example isnot biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of ΓC are, in contrast, noncommutative.

Embeddings of Danielewski surfaces

A Danielewski surface is defined by a polynomial of the form P=xnz−p(y). Define also the polynomial P′=xnz−r(x)p(y) where r(x) is a non-constant polynomial of degree ≤n−1 and r(0)=1. We show that, when n≥2 and deg p(y)≥2, the general fibers of P and P′ are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ℂ3. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(ydn>−1) for an infinite sequence of integers dn. We then consider a certain algebraic action of the orthogonal group \(\mathcal O(2)\) on ℂ4 which was first considered by Schwarz and then studied by Masuda and Petrie, who proved that this action could not be linearized. This was done by comparing the strata of this action to those of the induced tangent space action. Inequivalent embeddings of a certain singular Danielewski surface S in ℂ3 are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in ℂ3 which are algebraically non-isomorphic but holomorphically isomorphic.

A new example of a uniformly Levi degenerate hypersurface in C3

We present a homogeneous real analytic hypersurface in C3, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube ΓC over the two-dimensional real cone in R3 is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is not biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of ΓC are, in contrast, noncommutative.

Automorphisms of 2-Nondegenerate Hypersurfaces in ℂ 3

An exact upper estimate is obtained for the dimension of the automorphism group of a 2-dimensional hypersurface in ℂ3 possessing a Lie group structure.

On the dimension of a group transitively acting on a hypersurface in ℂ3

On the dimension of a group transitively acting on a hypersurface in ℂ3

Extension of smooth CR mappings between non-essentially finite hypersurfaces in C3

Extension of smooth CR mappings between non-essentially finite hypersurfaces in C3

ON THE LINEARIZATION OF AUTOMORPHISMS OF A REAL ANALYTIC HYPERSURFACE

It is proved that relative to certain normal coordinates any local automorphism of a nonspherical real-analytic hypersurface in Cn is a fractional-linear transformation. For a hypersurface in C3 a system of normal coordinates is constructed relative to which the entire stability group consists only of fractional-linear transformations. Bibliography: 13 titles.

Twistor CR manifolds and three-dimensional conformal geometry

A CR \text {CR} (i.e. partially complex) 5 5 -manifold is contructed as a sphere bundle over an arbitrary 3 3 -manifold with conformal metric. This so-called twistor C R CR manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract CR \text {CR} manifold to arise via the construction. The above correspondence is then used to prove that a twistor CR \text {CR} manifold is locally imbeddable as a real hypersurface in C 3 {{\mathbf {C}}^3} only if it is real-analytic with respect to a suitable atlas.

On the contact between complex manifolds and real hypersurfaces in 𝐶³

Let m m be a real C ∞ {\mathcal {C}^\infty } hypersurface of an open subset of C 3 {{\mathbf {C}}^3} and let p ∈ M p \in M . Let a 1 ( M , p ) {a^1}(M,p) denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of p p in C 3 {{\mathbf {C}}^3} with M M at p p . Let c 1 ( M , p ) {c^1}(M,p) denote the sup { m ∈ Z | \sup \{ m \in {\mathbf {Z}}| for all tangential holomorphic vector fields L L with L ( p ) ≠ 0 L(p) \ne 0 then L i 0 L ¯ j 0 … L i n L ¯ j n ( L M ( L ) ) ( p ) = 0 } {L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak {L}_M}(L))(p) = 0\} where i 0 , … , i n ; j 0 , … , j n {i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n} are positive integers such that ∑ t = 0 n i t + j t = m − 3 \sum \nolimits _{t = 0}^n {{i_t} + {j_t} = m - 3} and L M ( L ) {\mathfrak {L}_M}(L) denotes the Levi form of M M evaluated on the vector field L L . Theorem. If M M is pseudoconvex near p ∈ M p \in M then a 1 ( M , p ) = c 1 ( M , p ) {a^1}(M,p) = {c^1}(M,p) .