Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA characterization of CR quadrics with a symmetry property
111
0
0.0
(
0
)
Authors
Andrea Altomani
Ask ChatGPT about the research
No Arabic abstract
We study CR quadrics satisfying a symmetry property $(tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the $(tilde S)$ property in terms of their Levi-Tanaka algebras. In many cases the $(tilde S)$ property implies the $(S)$ property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric.
rate research
Read More
A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.
We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense maximally CR. One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory. Partial normal forms are obtained for general almost CR structures of any CR codimension, in particular, for almost-complex structures. Applications are given to the equivalence problem and the Lie group structure of the group of all CR-diffeomorphisms.
Applying Lies theory, we show that any $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in the class $mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at any point $p in M$ to be the $v$-axis in coordinates $(z, zeta, w = u + i, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $gamma$ is the $v$-axis so that $M = {u=F(z,zeta,overline{z},overline{zeta},v)}$ has Poincare-Moser reduced equation: begin{align} u & = zoverline{z} + tfrac{1}{2},overline{z}^2zeta + tfrac{1}{2},z^2overline{zeta} + zoverline{z}zetaoverline{zeta} + tfrac{1}{2},overline{z}^2zetazetaoverline{zeta} + tfrac{1}{2},z^2overline{zeta}zetaoverline{zeta} + zoverline{z}zetaoverline{zeta}zetaoverline{zeta} & + 2{rm Re} { z^3overline{zeta}^2 F_{3,0,0,2}(v) + zetaoverline{zeta} ( 3,{z}^2overline{z}overline{zeta} F_{3,0,0,2}(v) ) } & + 2{rm Re} { z^5overline{zeta} F_{5,0,0,1}(v) + z^4overline{zeta}^2 F_{4,0,0,2}(v) + z^3overline{z}^2overline{zeta} F_{3,0,2,1}(v) + z^3overline{z}overline{zeta}^2 F_{3,0,1,2}(v) + z^3{overline{zeta}}^3 F_{3,0,0,3}(v) } & + z^3overline{z}^3 {rm O}_{z,overline{z}}(1) + 2{rm Re} ( overline{z}^3zeta {rm O}_{z,zeta,overline{z}}(3) ) + zetaoverline{zeta}, {rm O}_{z,zeta,overline{z},overline{zeta}}(5). end{align} The values at the origin of Pocchiolas two primary invariants are: [ W_0 = 4overline{F_{3,0,0,2}(0)}, quadquad J_0 = 20, F_{5,0,0,1}(0). ] The proofs are detailed, accessible to non-experts. The computer-generated aspects (upcoming) have been reduced to a minimum.
In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.
A smooth, strongly $mathbb{C}$-convex, real hypersurface $S$ in $mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
