Point Of Infinite Type Research Articles

On the Growth of the Bergman Metric Near a Point of Infinite Type

We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints -- some of them necessary for a non-trivial Bergman space. However, these are mild constraints: unlike most earlier works on this subject, we are able to make estimates for non-convex pseudoconvex models as well. In fact, the domains we can analyse range from being mildly infinite-type to very flat at infinite-type boundary points.

A Note on Infinite Type Germs of a Real Hypersurface in

Abstract: The purpose of this article is to show that there exists a smooth real hypersurface germ of D'Angelo infinite type in such that it does not admit any (singular) holomorphic curve that has infinite order contact with at .
 2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25.
 Key words and phrases: Holomorphic vector field, automorphism group, real hypersurface, infinite type point.

On the automorphism group of a certain infinite type domain in [formula omitted

In this study, we consider an infinite type domain ΩP in C2. The aim of this study is to investigate the holomorphic vector fields tangent to an infinite type model in C2 vanishing at an infinite type point and to give an explicit description of the automorphism group of ΩP.

Gromov (non-)hyperbolicity of certain domains in ℂN

Abstract We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for 𝒞 1 , 1 ${\mathcal{C}^{1,1}}$ -smooth convex domains in ℂ 2 ${\mathbb{C}^{2}}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in ℂ n ${\mathbb{C}^{n}}$ are given.

Lower Bounds on the Kobayashi Metric Near a Point of Infinite Type

Under a potential-theoretical hypothesis named f-property which holds for all pseudoconvex domains of finite type and many examples of infinite type, we give a new method for constructing a family of bumping functions and hence plurisubharmonic peak functions with good estimates. The rate of lower bounds on the Kobayashi metric follows by the estimates of peak functions. The application to the continuous extendibility of proper holomorphic maps is given.

On the tangential holomorphic vector fields vanishing at an infinite type point

Let ( M , p ) (M,p) be a C ∞ \mathcal C^\infty smooth non-Leviflat CR hypersurface germ in C 2 \mathbb C^2 where p p is of infinite type. The purpose of this article is to investigate the holomorphic vector fields tangent to ( M , p ) (M,p) vanishing at p p .

Hypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite type

Hypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite type

Holomorphic Equivalence and Nonlinear Symmetries of Ruled Hypersurfaces in $\mathbb{C}^{2}$

We solve the local equivalence problem for ruled hypersurfaces in \(\mathbb {C}^{2}\) at a point of infinite type by giving a normal form. We describe all ruled hypersurfaces which admit nonlinear symmetries and give their complete characterization according to the dimension of their automorphism group. As an application, we give a bound for the dimension of the stability group for a family of nonminimal hypersurfaces.

The characterization of holomorphic vector fields vanishing at an infinite type point

This paper consider a rotationally symmetric real hypersurface in C 2 around the origin. We also assume that the origin is a point of infinite type in the sense of DʼAngelo. We prove that a holomorphic vector field defined on a neighborhood of origin which is tangent to the real hypersurface and vanishing at the origin is indeed a vector field generating rotational transformations.

Regularity of the [formula omitted]-Neumann problem at point of infinite type

We introduce general estimates for “gain of regularity” of solutions of the ∂ ¯ -Neumann problem and relate it to the existence of weights with large Levi form at the boundary. This enables us to discuss in a unified framework the classical results on fractional ellipticity (= subellipticity), superlogarithmic ellipticity and compactness. For each case, we exhibit a corresponding class of domains.

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points.

On the genericity of a bifurcation point of infinite type and multiple solutions of an elastic membrane

On the genericity of a bifurcation point of infinite type and multiple solutions of an elastic membrane

Critical point of infinite type in one dimension

An exact solution displaying a critical point of infinite type is given in one dimension. The properties are similar to those of a regular critical point, but with some quantitative differences.

Critical point of infinite type

We show that a critical point of infinite type is in fact a special case of first-order transition between two states: one perfectly ordered, and a disordered one. The analogy with exact models is pointed out.