We introduce new biholomorphic invariants for real-analytic hypersurfaces in 2-dimensional complex space and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained.
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.
In this article, we consider $\mathcal{C}^\infty$-smooth real hypersurfaces of infinite type in $\mathbb C^2$. The purpose of this paper is to give explicit descriptions for stability groups of the hypersurface $M(a,α,p,q)$ (see Sec. $1$) and a radially symmetric hypersurface in $\mathbb C^2$.
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 $1$.
We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, 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 $\mathbb C^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on …
The paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in $\mathbb{C}^2$. We also prove that, with the exception of hypersurfaces of the form $v = |z|^k$, local automorphisms are always determined by 1-jets. This proves a conjecture of D. Zaitsev in the finite type case.
We prove finite jet determination for (finitely) smooth CR diffeomorphisms of (finitely) smooth Levi degenerate hypersurfaces in $\mathbb{C}^{n+1}$ by constructing generalized stationary discs glued to such hypersurfaces.
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 show that every CR-automorphism of the closure of a Levi degenerate hyperquadric in the projective space extends to a holomorphic automorphism of the projective space.
We provide regularity results for CR-maps between real hypersurfaces in complex spaces of different dimension with a Levi-degenerate target. We address both the real-analytic and the smooth case. Our results allow immediate applications to the study of proper holomorphic maps between Bounded Symmetric Domains.
The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and analogous results for finite groups. The second part considers hypersurfaces of finite Catlin multitype and the Kohn-Nirenberg phenomenon in higher dimensions. We give a necessary condition for local convexifiability of a class of pseudoconvex hypersurfaces in $\mathbb C^{n+1}$.
We classify locally defined non-spherical real-analytic hypersurfaces in complex space whose Levi form has no more than one negative eigenvalue and for which the dimension of the group of local CR-automorphisms has the second largest value.
Let C(2,1) be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In a recent article, we proved that the CR-structures in C(2,1) are reducible to so(3,2)-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in C^3 that belong to C(2,1) and whose CR-curvature identically vanishes. Every such hypersurface is shown to be affinely equivalent to an open subset of the tube over the future light cone.
In our recent article (to appear in the Journal of Differential Geometry in 2016) we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we discovered that for the CR-curvature of such a hypersurface to vanish it suffices to require that only two coefficients (called $Θ^2_{21}$ and $Θ^2_{10}$) in the expansion of a single component of the CR-curvature form be identically zero. In this paper, we show that, surprisingly, the vanishing of the entire CR-curvature is in fact implied by the vanishing of a single quantity deri…
Recently, I. Kossovskiy and R. Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in ${\mathbb C}^2$ via the dimension of the algebra of infinitesimal automorphisms. In this note, we propose a short argument for obtaining their result.
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…
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.
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 obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniqueness of biholomorphisms with given 1-jet at some convenient point.
In our earlier articles we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper we provide an alternative shorter derivation of this equation by utilizing two invariants discovered by S. Pocchiola. We also investigate Pocchiola's invariants in the rigid case and give a partial classification of rigid 2-nondegenerate uniformly Levi degenerate of rank 1 hyper…
Let M be a connected real-analytic hypersurface in N-dimensional complex euclidean space whose Levi form is nondegenerate at some point. We prove that for every point p in M, there exists an integer k=k(M,p) such that germs at p of local real-analytic CR automorphisms of M are uniquely determined by their k-jets (at p). To prove this result we develop a new technique that can be seen as a resolution of the degeneracies of M. This procedure consists of blowing up M near an arbitrary point p in M regardless of its minimality or nonminimality; then, thanks to the blow-up, the original problem can…
We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.
In our recent work [25] we showed that $C^\infty$ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in $\mathbb C^{2}$ are not analytic in general. This result raised again the question on the nature of CR-maps of real-analytic hypersurfaces. In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the {\em sectorial analyticity property}. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeare…
We classify polynomial models for real hypersurfaces in $\mathbb C^N$, which admit nonlinearizable infinitesimal CR automorphisms. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some complex space.
Holomorphically homogeneous 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…
We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, the answer is expressed in terms of solutions of several well-known differential equations, in particular, the equation characterizing conformal metrics with constant negative curvature and a nonlinear $\bar\partial$-equation.
In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of an universal polynomial in the $7$-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely th…
We construct normal forms for Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. As one consequence, an explicit solution to the problem of local biholomorphic equivalence is obtained. Another consequence determines the dimension of the stability group of the hypersurface.
We consider hypersurfaces of finite type in a direct product space ${\mathbb R}^2 \times {\mathbb R}^2$, which are analogues to real hypersurfaces of finite type in ${\mathbb C}^2$. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in ${\mathbb C}^2$. For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the correspon…
We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with th…
For any real-analytic hypersurface M in complex euclidean space of dimension >= 2 which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p in M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their l(p)-jets at p. As a direct application, we derive a Lie group structure for the topological group Aut(M,p). Furthermore, we also show that the order l(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. As a first consequence, it follows that th…
We construct a complete convergent normal form for a real hypersurface in $\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. In particular, we obtain, in the spirit of the work of Chern and Moser \cite{chern}, distinguished curves in the Levi degeneracy set, that we call \it degenerate chains.
We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. 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 $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint w…
In this paper, we consider real hypersurfaces $M$ in $\Bbb C^3$ (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require the hypersurface to satisfy a certain second order nondegeneracy condition (called 2-nondegeneracy) at the point. Our first result is the construction of a principal bundle $P\to M$ with an absolute parallelism, uniquely determined by the CR structure on $M$, which reduces the question of whether two such CR manifolds…
The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. The results give for all such hypersurfaces a complete normalization which respects the symmetries. In particular, they apply to tubes and rigid hypersurfaces, providing an effective classification. The main tool is a complete normal form constructed for a general hypersurface with a tube model. As an application, we describe all bi…
We establish an injective correspondence $M\longrightarrow\mathcal E(M)$ between real-analytic nonminimal hypersurfaces $M\subset\mathbb{C}^{2}$, spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound $\mbox{dim}\,\mathfrak{hol}(M,p)\leq 5$ for the infinitesimal automorphism algebra of an \it arbitrary \rm germ $(M,p)\not\sim(S^3,p')$ of a real-analytic Levi nonflat hypersurface $M\subset\mathbb{C}^2$ (the Dimension Conjecture). This bound gives the first proof of the dimension gap $\mbox{dim}\,\math…
We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in $\mathbb{C}^{n+1}$, which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class $\mathcal{C}^4$. This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.
Let ${\mathfrak C}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. We show that the CR-structures in ${\mathfrak C}_{2,1}$ are reducible to ${\mathfrak{so}}(3,2)$-valued absolute parallelisms and give applications of this result.
Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold.
We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every sta…
