We investigate CR-manifolds which are tubes M:= F x iV over general bases F in a real vector space V and characterize the k-nondegeneracy of M in terms of the real affine geometry of F. We give a method for an explicit computation of the Lie algebra hol(M,a) of all local infinitesimal CR-transformations at a and use these local invariants to establish the CR-inequivalence of certain families of CR-manifolds. In dimension 5 we present, apart from the well known tube over the future light cone, new examples of 2-nondegenerate homogeneous CR-manifolds that are mutually locally CR-inequivalent.
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.
In this paper we studied a broader type of generalized balls which are domains on the complex projective with possibly Levi-degenerate boundaries. We proved rigidity theorems for proper holomorphic mappings among them by exploring the structure of the moduli spaces of projective linear subspaces, which generalized some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries.
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.
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.
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.
In this paper we present new examples of homogeneous 2-nondegenerate CR-manifolds of dimension 5 and give, up to local CR-equivalence, a full classification of all CR-manifolds of this type.
Let $S$ be a real-analytic hypersurface in $C^n$ or, more generally, a CR-submanifold in a complex manifold $Z$ of arbitrary CR-codimension. This paper addresses the question, when a local biholomorphic map between open sets in $Z$ sending an open piece of $S$ into $S$ extends to a global biholomorphic self-map of $Z$ preserving $S$. We give answers for a large class of Levi-degenerate group orbits in compact Hermitian symmetric spaces.
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.
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 $\mathbb C^{n+1}$.
Homogeneous complex manifolds satisfying various types of Levi conditions are considered. Classical results which were of particular interest to Andreotti are recalled. Convexity and concavity properties of flag domains are discussed in some detail. A precise classification of pseudoconvex flag domains is given. It is shown that flag domains which are in a certain sense generic are pseudoconcave.
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 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…
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…
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.
Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $π:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong π^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.
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…
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.
Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M is greater than or equal to 5 and if dim M = 5, then k= 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exists a canonical Cartan connection, modelled on a suitable projective completion of the tube over the future light cone {z\in C^3: (x^1)^2+(x^2)^2-(x^3)^2 = 0, x^3>0}. This determines a complete solution to the equivalence problem for this class of CR manifolds.
We prove that for every $n\geq 3$ the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, $(2n+1)$-dimensional CR manifolds of hypersurface type with a $1$-dimensional Levi kernel is equal to $n^2+7$, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka's conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.
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.
In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for $M$. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M.
We introduce new invariant tensors in CR structures which can be viewed as higher order Levi forms. Using the second and third order tensors, we give a complete formal normal form (in the sense of Chern-Moser) for a real hypersurface at a generic Levi degeneracy. (We say that M has a generic Levi degeneracy at p if the Levi determinant vanishes at p but its differential does not, and the set of Levi degenerate points of M is transverse to the Levi null space at p.) By applying a convergence theorem for formal mappings due to the author, Baouendi, and Rothschild, we conclude that the above ment…
We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasi-homogeneous polynomials with isolated singularity. This result generalizes previous results of Burns-Gong and Fernández-Pérez. Furthermore, we prove the existence of two new rigid normal forms for singular real-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume.
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 …
We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core at $x\in M$. It consists of a $\mathbb Z$-graded vector space $\mathfrak{m}_x=\bigoplus_{p\leq k-2}\mathfrak{m}_x^p$ of height $k-2$ endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of $(M,\mathcal D,\mathcal J)$ and the Levi forms of highe…
Let $F(z)=\mathcal{R}e(P(z)) + h.o.t$ be such that $M=(F=0)$ defines a germ of real analytic Levi-flat at $0\in\mathbb{C}^{n}$, $n\geq{2}$, where $P(z)$ is a homogeneous polynomial of degree $k$ with an isolated singularity at $0\in\mathbb{C}^{n}$ and Milnor number $μ$. We prove that there exists a holomorphic change of coordinate $ϕ$ such that $ϕ(M)=(\mathcal{R}e(h)=0)$ where $h(z)$ is a polynomial of degree $μ+1$ and $j^{k}_{0}(h)=P$.
We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a …
Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive $\ddbar$-closed currents T of bidimension (1,1) and of mass 1 directed by the directions where all continuous psh functions in $U$ have vanishing Levi-form. The extremal ones, are supported on the sets where all continuous psh functions are con…
This article is dedicated to the centenary of the local CR equivalence problem, formulated by Henri Poincaré in 1907. The first part gives an account of Poincaré's heuristic counting arguments, suggesting existence of infinitely many local CR invariants. Then we sketch the beautiful completion of Poincaré's approach to the problem in the work of Chern and Moser on Levi nondegenerate hypersurfaces. The last part is an overview of recent progress in solving the problem on Levi degenerate manifolds.
Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension $(2n+1)$ have maximal symmetry dimension $n^2+4n+3$. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is $n^2+4$ for Levi-indefinite structures and $n^2+3$ for Levi-definite structures when $n>1$. In the exceptional case of CR-dimension $n=1$, the submaximal symmetry dimension 3 was computed by E.\,Cartan.
We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group $G(M)$ of automorphisms of $M$. We characterize also the standard homogeneous CR manifolds as the homogeneous CR manifolds whose group $G(M)$ in not semisimple.
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 classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.
Applying Lie's theory, we show that any $\mathcal{C}^ω$ 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 $(z, ζ, w = u + i\, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $γ$ is the $v$-axis so that $M = \{u=F(z,ζ,\overline{z},\overlineζ,v)\}$ has Poincaré-Moser reduced equation: \begin{align} u & = z\overline{z} + \…
We construct several new examples of homogeneous domains in complex space that do not have bounded realisations. They are equivalent to tubes over affinely homogeneous domains in real space and have a real-analytic everywhere Levi non-degenerate non-umbilic boundary. Using the geometry of the boundary, we determine the full automorphism groups of the domains. We also discuss some interesting examples of tube domains with everywhere umbilic boundary (i.e., boundary equivalent to the corresponding quadric).
Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every point a in M the Lie algebra hol(M,a) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields and the Euler vector field. Under these assumptions we show that: (I) every hol(M,a) consists of polynomial vector fields, hence coincides with the Lie algebra hol(M) of all infinitesimal real-analytic CR-automorphisms of M; (II) every local real-analytic CR-automorphism of M ex…
Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integr…
Germs of locally homogeneous CR manifolds M can be characterized in terms of certain algebraic data, e.g., by CR-algebras. We give an explicit formula which relates the Levi form of such an M and its higher order analogues to the Lie brackets in certain finite dimensional Lie algebras. As an application we give a simple characterization of geometric properties of M such as minimality, k-nondegeneracy, holomorphic degeneracy etc. in purely algebraic terms. We present an example of a homogeneous 3-nondegenerate CR-manifold. We also determine a universal upper bound for the order k of k-nondegene…
We construct canonical absolute parallelisms over real-analytic manifolds equipped with $2$-nondegenerate, hypersurface-type CR structures of arbitrary odd dimension not less than $7$ whose Levi kernel has constant rank belonging to a broad subclass of CR structures that we label as recoverable. For this we develop a new approach based on a reduction to a special flag structure, called the dynamical Legendrian contact structure, on the leaf space of the CR structure's associated Levi foliation. This extends the results of Porter-Zelenko [20] from the case of regular CR symbols constituting a d…
We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a positive answer to a question of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Ampère equation and we obtain a generalization the work of P.M. Wong on the classification of complete weighted circular domains.
In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.
An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associated solvable Lie algebra, up-to-coverings. Moreover, for homogeneous $(M^4,J)$ we provide examples with connected and non-connected double covering, thus proving that in general the double absolute parallelism is not the restriction of two absolute parallelisms. Furthermore, it …
Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions $\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under split-diffeomorphisms $(x,y,a,b) \,\longmapsto\, \big( f(x,y),\,g(x,y),\,φ(a,b),\,ψ(a,b) \big)$. Two Levi forms exist, and have the same rank $r \leqslant \min (n,m)$. If $\mathcal{M}$ is $k$-nondegenerate with respect to parameters and $l$-nondegenerate with respect to variables, $\mbox{Aut}(\mathcal{M})$ is a local Lie group of dimension: \[ \dim\, \mbox{Aut} (\mathcal{M}) …
It is shown that two Levi-Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by Beloshapka.
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.
Let $(X,ω)$ be an $n$-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form $(ω+dd^cφ)^m\wedge ω^{n-m}=F(x,φ)ω^n.$ Under some natural conditions on $F$, this equation has a unique continuous solution. When $(X,ω)$ is rational homogeneous we further show that the solution is Hölder continuous.
