Atsushi Hayashimoto, Ninh Van Thu
The purpose of this paper is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in $\mathbb C^2$. The decompositions of infinitesimal CR automorphisms are also given.
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We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if $X$ is such an infinitesimal CR automorphism, then $λX$ is smooth for all multipliers $λ$. As an application, we study the regularity of infinitesimal automorphisms of certain embedded hypersurfaces of infinite type.
We determine infinitesimal $\mathrm{CR}$ automorphisms and stability groups of real hypersurfaces in $\mathbb C^2$ in the case when the hypersurface is nonminimal and of infinite type at the reference point.
We determine the Lie algebras of infinitesimal automorphisms for the models of the CR-manifolds belonging to general classes ${\sf II}$, ${\sf III_2}$ and ${\sf IV_2}$ through Cartan's equivalence method.
We consider a problem whether a CR mapping of a generic manifold in complex space is uniquely determined by its finite jet at a point, which is referred to as finite jet determination. We derive the finite jet determination for CR mappings of smooth Levi nondegenerate manifolds of arbitrary codimension from the finite dimensionality of the algebras of infinitesimal automorphisms of the corresponding quadrics. Previously, this implication was known for real analytic manifolds. We prove a new 2-jet determination result that covers most affirmative results on this matter obtained so far.
We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case…
In this article, we consider a real smooth hypersurface $M\subset \mathbb C^2$, which is of infinite type at $p\in M$. The purpose of this paper is to show that the real vector space of tangential holomorphic vector field germs at $p$ vanishing at $p$ is either trivial or of real dimension $1$.
Regularity of infinitesimal automorphisms of involutive structures
本文证明了对合结构的无穷小自同构是光滑的，通过构建$V$-联络的正则性理论，证明$V$-截面满足Hans Lewy定理的类比形式，并引入确保光滑性的非退化条件。
Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$ having a point of Levi-nondegeneracy. The following alternative is demonstrated for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote by $d$ the dimension of $s$ or $G$. Then (i) either $d=n^2+4n+3$ and $M$ is spherical everywhere; (ii) or $d\le n^2+2n+2+δ_{2,n}$ and in the case of equality $M$ is spherical of fixed Levi signature in the open dense subset of Levi-nondegenerate points. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (…
We investigate the space of abelian relations of planar webs admitting infinitesimal automorphisms. As an application, we construct 4k-14 new algebraic families of global exceptionnal k-webs on the projective plane, for each k >4.
We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new "secondary" gap phenomenon.
A holomorphic mapping $H$ between two real-analytic CR manifolds $M$ and $M'$ is said to be locally rigid if any other holomorphic map $F\colon M \to M'$ which is close enough to $H$ is obtained by composing $H$ with suitable automorphisms of $M$ and $M'$. With the aim of reducing the local rigidity problem to a linear one, we provide sufficient infinitesimal conditions. Furthermore we study some topological properties of the action of the automorphism group on the space of nondegenerate mappings from $M$ to $M'$.
Characterization of real-analytic infinitesimal CR automorphisms for a class of hypersurfaces in $\Bbb C^4.$
本文刻画了由齐次多项式定义的 $\Bbb C^4$ 中模型超曲面的实解析无穷小 CR 自同构李代数，分类了单项式情形下的幂零旋转，并给出广义旋转实维数为 3 的例子。
We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de-Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the is well know complex for locally conformally flat manifolds. Recently, a theory of subcomplexes in BGG sequences has been developed. This appl…
In this paper, we prove that if $X$ is a Riemann surface of infinite analytic type and $[μ]_T$ is any element of Teichmüller space, then there exists $μ_1 \in [μ]_T$ so that $μ_1$ is an infinitesimal Strebel point.
We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb D$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning.vWe also give a new proof of the classical Cowen-Pommerenke inequalities for univalent self-maps of $\mathbb D$.
Infinitesimal bendings for classes of two-dimensional surfaces in $\mathbb{R}^3$ are investigated. The techniques used to construct the bending fields include reduction to solvability of Bers-Vekua type equations and systems of differential equations with periodic coefficients.
We continue the development of the theory of infinitesimal Lipschitz equivalence, showing the genericity of the condition for families of hypersurfaces with isolated singularities.
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