Hypersurfaces Of Finite Type Research Articles

Spectral stability of the Kohn Laplacian under perturbations of the boundary

In this paper, we study stability of the spectrum of the Kohn Laplacian □b on the boundary of a smoothly bounded domain in Cn as the boundary is perturbed in the C2-topology. We obtain estimates for spectral stability of the Kohn Laplacian on smooth compact hypersurfaces that satisfy uniform subelliptic estimate, in particular for strictly pseudoconvex hypersurfaces in Cn and pseudoconvex hypersurfaces of finite type in C2.

Normal forms of para-CR hypersurfaces

We consider hypersurfaces of finite type in a direct product space R2×R2, which are analogues to real hypersurfaces of finite type in C2. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in C2. 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 corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted ℓ-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.

Rigid hypersurfaces and infinitesimal CR automorphisms

We study a survey on the relations between rigid hypersurfaces and infinitesimal CR automorphisms. After reviewing the case of hypersurfaces of finite type, we study the case of hypersurfaces of infinite type. Some open problems are posed in the last section.

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.

Convergent normal form and canonical connection for hypersurfaces of finite type in [formula omitted

We study the holomorphic equivalence problem for finite type hypersurfaces in C2. We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in C2.

$L_k$-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES

In this article, we study $L_k$-finite-type hypersurfaces $M^n$ of a hyperbolic space $\mathbb{H}^{n+1}\subset\mathbb{R}^{n+2}_1$, for $k\geq 1$. In the 3-dimensional case, we obtain the following classification result. Let $\psi:M^3\rightarrow\mathbb{H}^{4}\subset\mathbb{R}^5_1$ be an orientable hypersurface with constant $k$-th mean curvature $H_k$, which is not totally umbilical. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is an open portion of a standard Riemannian product $\mathbb{H}^1(r_1)\times\mathbb{S}^{2}(r_2)$ or $\mathbb{H}^2(r_1)\times\mathbb{S}^{1}(r_2)$, with $-r_1^2+r_2^2=-1$. In the $n$-dimensional case, we show that a hypersurface $M^n\subset\mathbb{H}^{n+1}$, with constant $k$-th mean curvature $H_k$ and having at most two distinct principal curvatures, is of $L_k$-2-type if and only if $M^n$ is an open portion of a Riemannian product $\mathbb{H}^m(r_1)\times\mathbb{S}^{n-m}(r_2)$, with $-r_1^2+r_2^2=-1$, for some integer $m\in\{1,\dots,n-1\}$. In the case $k=n-1$ we drop the condition on the principal curvatures of the hypersurface $M^n$, and prove that if $M^n\subset\mathbb{H}^{n+1}$ is an orientable $H_{n-1}$-hypersurface of $L_{n-1}$-2-type then its Gauss-Kronecker curvature $H_{n}$ is a nonzero constant.

Stationary Discs for Smooth Hypersurfaces of Finite Type and Finite Jet Determination

We construct a finitely dimensional manifold of invariant holomorphic discs attached to a certain class of smooth pseudoconvex hypersurfaces of finite type, generalizing the notion of stationary discs. The discs we construct are determined by a finite jet at a given boundary point and their centers fill an open set. As a consequence, we obtain finite jet determination of CR diffeomorphisms between smooth hypersurfaces of this class.

Non-embeddable real algebraic hypersurfaces

We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.

Normal forms and symmetries of real hypersurfaces of finite type in $\\mathbb C^2$

We give a complete description of normal forms for real hypersurfaces of finite type in C-2 with respect to their holomorphic symmetry algebras. The normal forms include refined versions of the constructions by Chern-Moser, Stanton, Kolar. We use the method of simultaneous normalisation of the equations and symmetries that goes back to Lie and Cartan. Our approach leads to a unique canonical equation of the hypersurface for every type of its symmetry algebra. Moreover, even in the Levi-degenerate case, our construction implies convergence of the transformation to the normal form if the dimension of the symmetry algebra is at least two. We illustrate our results by explicitly normalising Cartan's homogeneous hypersurfaces and their automorphisms.

Finite type hypersurfaces with divergent normal form

We consider the convergence problem for normal forms on finite type hypersurfaces in dimension two, constructed in Kolař (Math Res Lett 12:897–910, 2005). Two families of examples of hypersurfaces with divergent normal form are given. The results also show that the divergence phenomenon cannot be avoided by a modification of the normal form construction.

Local equivalence of symmetric hypersurfaces in~$\mathbb C^2$

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 complete normalizations for such hypersurfaces, which respect 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 biholomorphic maps between tubes, answering a question posed by N. Hanges. Similar results for hypersurfaces admitting nontransversal symmetries are obtained.

Degenerate hypersurfaces with a two-parametric family of automorphisms

We give a complete classification of Levi-degenerate hypersurfaces of finite type in ℂ2 with two-dimensional symmetry groups. Our analysis is based on the classification of two-dimensional Lie algebras and an explicit description of isotropy groups for such hypersurfaces, which follows from the construction of Chern–Moser type normal forms at points of finite type, developed in [M. Kolář, Normal forms for hypersurfaces of finite type in ℂ2, Math. Res. Lett. 12 (2005), pp. 897–910].

Degenerate real hypersurfaces in ℂ² with few automorphisms

We introduce new biholomorphic invariants for real-analytic hypersurfaces in C 2 \mathbb {C}^2 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.

Finite type Monge–Ampère foliations

Abstract For plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation whose level sets are hypersurfaces of finite type, in dimension 2, it is shown that the Monge–Ampère foliation is defined even at points of higher degeneracy. The result is applied to provide a positive answer to a question of Burns on homogeneous polynomials whose logarithms satisfy the complex Monge–Ampère equation and to generalize the work of P. M. Wong on the classification of complete weighted circular domains.

Generalized models and local invariants of Kohn–Nirenberg domains

The main obstruction for constructing holomorphic reproducing kernels of Cauchy type on weakly pseudoconvex domains is the Kohn–Nirenberg phenomenon, i.e. nonexistence of supporting functions and local nonconvexifiability. In this paper we give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.

Local symmetries of finite type hypersurfaces in ℂ2

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in ℂ2. It is also proved that, with the exception of hypersurfaces of the form v = |z| k , local automorphisms are always determined by their 1-jets. Using this result, the second part describes special normal forms which by an additional normalization eliminate the nonlinear symmetries of the model and allows to decide effectively about local equivalence of two hypersurfaces given in this normal form.

Boundary regularity of correspondences in ℂn

LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂn and\(\hat f\) a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that\(\hat f\) must extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich-Pinchuk extension result for CR maps.

Normal forms for hypersurfaces of finite type in $\mathbb{C}^2$

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.

Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$2 at a point p ∈ M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case. If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$3. We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$2.

On averaging operators associated with convex hypersurfaces of finite type

On averaging operators associated with convex hypersurfaces of finite type