Q-Complete Domains with Corners and Extension of Line Bundles
Final version to appear in Math.Z
math.CV↗
arXiv subjects
John Erik Fornaess
Publications and source records attributed to John Erik Fornaess.
Solving $\bar\partial_b$ on hyperbolic laminations
Let $X$ denote a compact set which is laminated by Riemann surfaces. We assume that $X$ carries a positive CR line bundle $ L\rightarrow X$. The main result of the paper is that there exists a positive integer $s$ so that if $v$ is any continuous $(0,1)$ form with coefficients in $L^{\otimes s}$ there exists a continuous section $u$ of $L^{\otimes s}$ solving the equation $\bar\partial_b u=v$.
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Examples of Minimal Laminations and Associated Currents
In this paper, we construct various examples of holomorphic laminations, with leaves of dimension 1, and we also study some of their dynamical properties. In particular we study existence and uniqueness of positive closed currents. We construct minimal laminations with infinitely many mutually singular closed currents and no non-closed harmonic current. We also consider embeddings to projective space.
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On supnorm estimates for $\bar\partial$ on infinite type convex domains in $\mathbb C^2$
In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.
math.CV↗
Asymptotic behavior of the Kobayashi metric in the normal direction
In this paper, we construct a pseudoconvex domain in $\mathbb C^3$ where the Kobayashi metric does not blow up at a rate of one over distance to the boundary in the normal direction.
math.CV↗
Kobayashi, Carathéodory, and Sibony metric
In this paper, we estimate the boundary behavior of the Sibony metric near a pseudoconcave boundary point. We show that the metric blows up at a different rate than the Kobayashi metric.
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Infinite type germs of smooth hypersurfaces in $\mathbb C^n$
In this paper, we discuss germs of smooth hypersurface in $\mathbb C^n$. We show that if a point on the boundary has infinite D'Angelo type, then there exists a formal complex curve in the hypersurface through that point.
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Unique Ergodicity of Harmonic Currents on Singular Foliations of P2
Let F be a holomorphic foliation of P^2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a unique positive harmonic $(1,1)$ current $T$ of mass one, directed by F. This implies strong ergodic properties for the foliation. We also study the harmonic flow associated to the current $T.$
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Riemann Surface Laminations with Singularities
In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in CP^2 and for generic holomorphic foliations in CP^2.
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A note on plurisubharmonic defining functions in C^2
Let D be a smoothly bounded domain in C^2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
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The Julia set for Henon maps
In this paper we investigate the support of the unique measure of maximal entropy of complex Henon maps, J^*. The main question is whether this set is the same as the analogue of the Julia set, J.
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Local $L^2$ results for $\bar\partial$: The isolated singularities case
Let $X$ be a pure n-dimensional complex analytic set in $\mathbb{C}^N$ with an isolated singularity at 0. We study the Cauchy-Riemann operator on a deleted neighborhood of the singular point 0 in $X$.
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Semiglobal results for $\bar\partial$ on a complex space with arbitrary singularities
We obtain some $L^2$ results for the Cauchy-Riemann operator on forms that vanish to high order near the singular set of a complex space.
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The Several Complex Variables Problem List
The purpose of this bulletin board is to collect problems in higher dimensional complex analysis. We are interested both in basic research questions as well as interactive questions with other fields and sciences. We encourage everybody to submit problems to the list. This includes not only those coming up in your own work, but also others- maybe well known and classical- that you see missing, but that you think workers in the field should be aware of. Not only are we searching for basic research type questions in several complex variables, we also solicit questions exploring relations to other mathematical fields, one complex variable, partial differential equations, differential geometry, dynamics, etc. and to other sciences such as physics, engineering, biology etc. While some questions fall rather naturally into one of the subject areas in this problem list and may lead to a publishable paper, other questions may be non- specific or of a transient or technical quality. Thus we have a section called ``Scratchpad'' for conversational questions, vaguely formulated questions, or questions to which you may hope to get a quick answer. In the ``open prize problems'' section, you are welcome to offer a nice little prize to whoever does it. The miscellaneous section is for announcements of conferences, jokes, remarks on the general state of the field, etc.
math.CV↗
An embedding of {\bf C} in {\bf C}$^2$ with hyperbolic complement
Let $X$ be a closed, $1$-dimensional, complex subvariety of $\CC^2$ and let $\ol{\BB}$ be a closed ball in $\CC^2 - X$. Then there exists a Fatou-Bieberbach domain $Ω$ with $X \subseteq Ω\subseteq \CC^2 - \ol{\BB}$ and a biholomorphic map $Φ: Ω\ra \CC^2$ such that $\CC^2 - Φ(X)$ is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in $\CC^2$ whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.
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Compositional roots of Hénon maps
Let $H$ denote a composition of complex Hénon maps in {\bf C}$^2$. In this paper we show that the only possible compositional roots of $H$ are also compositions of Hénon maps, and that $H$ can have compositional roots of only finitely many distinct orders.
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