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In this paper, we find surprisingly small Oka domains in Euclidean spaces ${\mathbb {C}}^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in ${\mathbb {C}}^n$, we show that ${\mathbb {C}}^n\setminus E$ is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces $\Sigma _t\subset {\mathbb {C}}^n$ for $t\in {\mathbb {R}}$ dividing ${\mathbb {C}}^n$ in an unbounded hyperbolic domain and an Oka domain such that at $t=0$, $\Sigma _0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in ${\mathbb {C}}^n$ for $n>1$ whose projective closure $\overline E\subset \mathbb {C}\mathbb {P}^n$ avoids a hyperplane $\Lambda \subset \mathbb {C}\mathbb {P}^n$ and is polynomially convex in $\mathbb {C}\mathbb {P}^n\setminus \Lambda \cong {\mathbb {C}}^n$, then ${\mathbb {C}}^n\setminus E$ is an Oka domain.

We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of Yuta Kusakabe to the effect that the complement $\mathbb C^n\setminus K$ of any polynomially convex subset $K$ of $\mathbb C^n$ is an Oka manifold. The analogous result is obtained with $\mathbb C^n$ replaced by any Stein manifold with the density property., A few minor corrections were made. To appear in Math. Res. Lett

In this paper we survey the theory of holomorphic approximation, from the classical nineteenth century results of Runge and Weierstrass, continuing with the twentieth century work of Oka and Weil, Mergelyan, Vitushkin, and others, to the most recent ones on higher dimensional manifolds. The paper includes some new results and applications of this theory, especially to manifold-valued maps.

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.

We prove that a Stein manifold of dimension $d$ admits a proper holomorphic embedding into any Stein manifold of dimension at least $2d+1$ satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan pertaining to embeddings into complex Euclidean spaces, as well as several other recent results., To appear in J. d'Analyse Math

We prove that a non-autonomous basin with bounds is an Oka manifold. A consequence is that it has an abundance of holomorphic maps from C m \mathbb {C}^m into it, and in particular it does not carry a non-constant bounded plurisubharmonic function.

We construct a holomorphic embedding \(\phi :\mathbb B^3\rightarrow {\mathbb {C}}^3\) such that \(\phi ({\mathbb {B}}^3)\) is not Runge in any strictly larger domain. As a consequence, \(\mathcal S\ne {\mathcal {S}}^1\) for \(n=3\).

We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure., Misprints corrected, added Sublemma 3.4 to the proof of Lemma 3.1

We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over finite dimensional locally finite CW-complexes. This generalizes work of Thurston–Vaserstein and Vaserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Andersen-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases. However, we disprove the degree bound for \(3\)-jets in dimension 4. This follows from the fact that if \(\Sigma \) is the disjoint union of \(r=7\) involutive lines in \(\mathbb P^3\), then \(\Sigma \) is contained in a degree \(d=4\) hypersurface, i.e., the restriction morphism \(\iota :H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(4))\rightarrow H^0(\Sigma ,\mathcal O_{\Sigma }(4))\) has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if \((r,d)\ne (7,4)\) then the map \(\iota \) has maximal rank.

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in Cn. For a bounded strongly pseudoconvex domain in Cn and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to the identity map with respect to the C1 -norm and maps the boundary point to a strongly convex boundary point. © 2018. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: https://doi.org/10.1090/proc/13693

We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.

We prove that for a strongly pseudoconvex domain $$D\subset \mathbb {C}^n$$ , the infinitesimal Caratheodory metric $$g_C(z,v)$$ and the infinitesimal Kobayashi metric $$g_K(z,v)$$ coincide if z is sufficiently close to bD and if v is sufficiently close to being tangential to bD. Also, we show that every two close points of D sufficiently close to the boundary and whose difference is almost tangential to bD can be joined by a (unique up to reparameterization) complex geodesic of D which is also a holomorphic retract of D. The same continues to hold if D is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed; this has consequences for the behavior of the squeezing function.

In this paper we give a very simple proof of the existence and plenitude of Runge tubes in $\mathbb C^n$ $(n>1)$ and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold $A$ of codimension at least two in a complex Euclidean space $\mathbb C^n$, the normal bundle of $A$ in $\mathbb C^n$ admits a holomorphic embedding onto a Runge domain in $\mathbb C^n$ which agrees with the inclusion map $A\hookrightarrow \mathbb C^n$ on the zero section., In this version we added Remark 1.6 and Problem 1.7. The paper will appear in Proc. Amer. Math. Soc

In recent work by Zimmer it was proved that if Ω ⊂ C n is a bounded convex domain with C ∞-smooth boundary, then Ω is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if Ω is only assumed to be C 2 –smooth. © 2018. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: https://doi.org/10.2140/pjm.2018.297.79

We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of ℂ n \mathbb{C}^{n} .

We show the existence of free dense subgroups, generated by 2 elements, in the holomorphic shear and overshear group of complex-Euklidean space and extend this result to the group of holomorphic automorphisms of Stein manifolds with Density Property, provided there exists a generalized translation. The conjugation operator associated to this generalized translation is hypercyclic on the topological space of holomorphic automorphisms., 9 pages

We construct knotted proper holomorphic embeddings of the unit disc in

In this paper we prove the equivalence of two definitions of laminated currents.

We show that for any bounded domain \(\varOmega\subset\mathbb{C} ^{n}\) of 1-type 2k which is locally convexifiable at p∈bΩ, having a Stein neighborhood basis, there is a biholomorphic map \(f:\bar{\varOmega}\rightarrow\mathbb{C} ^{n} \) such that f(p) is a global extreme point of type 2k for \(f{(\overline{\varOmega})}\).

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in C in terms of the asymptotic behaviour of quotients of invariant metrics. The final version of this research has been published in Matematische Zeitschrift. © 2017 Springer Verlag

We study conformal structure and topology of leaves of singular foliations by Riemann surfaces. The final version of this research has been published in the Journal of Geometric Analysis. © 2017 Springer Verlag

We study ellipticity properties of complements of compact subsets of Stein manifolds., Comment: 18 pages, 2 figures

We show that if a compact set X in \({\mathbb P^n}\) is laminated by holomorphic submanifolds of dimension q, then \({\mathbb P^n{\setminus}X}\) is (q + 1)-complete with corners. Consider a manifold U, q-complete with corners. Let \({\mathcal N}\) be a holomorphic line bundle in the complement of a compact in U. We study when \({\mathcal N}\) extends as a holomorphic line bundle in U. We give applications to the non existence of some Levi-flat foliations in open sets in \({\mathbb P^n}\). The results apply in particular when U is a Stein manifold of dimension n ≥ 3, then every holomorphic line bundle in the complement of a compact extends holomorphically to U.

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\bar{\Omega})$ is that there is a holomorphic disk $D \subset \bar{\Omega}$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.

We give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class \({\mathcal C^k}\), k ≥ 1, by entire functions.

We will show that the currents of Duval and Sibony can be constructed easily from Poletsky disks.

We construct a 2-dimensional complex manifold X which is the increasing union of proper subdomains that are biholomorphic to ℂ2, but X is not Stein.

In this article we study generic properties of totally real submanifolds M of . On the one hand we show that if M is polynomially convex and also has bounded exhaustion hulls, then sufficiently small -perturbations of M are polynomially convex and have bounded exhaustion hulls. The proof of this relies on an explicit local function (the Gaussian kernel) and the solution of a Cousin I problem. On the other hand, if , there exist arbitrarily small -perturbations M′ of M such that M′ is polynomially convex and has bounded exhaustion hulls. The proof is a further development of an idea of Forstneric and Rosay to push different pieces of M into different complex submanifolds of . We also give a perturbation result that can be obtained by applying a holomorphic automorphism.

We study topological properties of attracting sets for automorphisms of ℂk. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

In this paper we discuss approximation of partially smooth functions. The problem arises naturally in the study of laminated currents.

We prove that for any complete pluripolar set $X\subset\C^2$ (in particular for any analytic subset) there exists a proper holomorphic embedding $\phi\colon\triangle\emb\C^2$ of the open unit disk $\triangle\subset\C$ such that $\phi(\triangle)\cap X=\emptyset$. It follows that the same holds true in $\C^n$ for any $n>1$.

Nous avons fait des progres sur le probleme du plongernent des surfaces de Riemann ouvertes dans C 2 . Il est connu que pour tout entier naturel d≥ 2, le nombre N d := [3d / 2] + 1 est le plus petit entier naturel pour lequel il existe un plongement propre de toute variete de Stein de dimension d dans C N d. Le probleme du plongement propre des varietes de Stein de dimension 1 dans C 2 reste ouvert (il existe du plongement propre dans C 3 ). Dans ce texte nous prouvons le resultat suivant: soit T un tore complexe de dimension 1; alors il existe un plongement propre de toute partie de T, dont la frontiere a un nombre fini de composantes (aucune d'elle n'etant un point), dans C 2 . Nous prouvons aussi que les algebres de fonctions analytiques sur certaines surfaces de Riemann sont doublement generees.

For certain bordered submanifolds M ⊂ ℂ2 we show that M can be embedded properly and holomorphically into ℂ2. An application is that any subset of a torus with two boundary components can be embedded properly into ℂ2.

We show that for any m ∈ ℕ ∪ {∞}, there exist m disjoint FB (Fatou–Bieberbach) domains whose union is dense in ℂk. In fact, we show that any point not in the union is a boundary point for all the domains. We construct FB domains that contains arbitrary countable collections of subvarieties of ℂk, and we construct FB domains that intersect elements of countable collections of affine subspaces of ℂk in connected proper subsets. Moreover, we show that any Runge FB domain is the attracting basin for a sequence of automorphisms of ℂk, although not necessarily if you only allow iteration of one automorphism. We also show that an increasing sequence of Runge ℂk's is a ℂk.

We show that any finitely connected domain Open image in new window can be properly embedded into Open image in new window. For some sequences Open image in new window can also be properly embedded into Open image in new window.

We give estimates for the squeezing function on strictly pseudoconvex domains, and derive some sharp estimates for the Caratheodory, Sibony and Azukawa metrics near their boundaries.

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold., Comment: Comments are welcome!

We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex., Slighlty revised version; accepted in Proc. Amer. Math. Soc

We prove that any Loewner PDE in a complete hyperbolic starlike domain of $\C^N$ (in particular in bounded convex domains) admits an essentially unique univalent solution with values in $\C^N$., Comment: 8 pages

We study bounded domains in [Formula: see text] with certain smoothness conditions and the properties of their squeezing functions in order to prove that the domains are biholomorphic to the ball.

We give a characterization of stratified totally real sets that admit Carleman approximation by entire functions. As an application we show that the product of two stratified totally real Carleman sets is a Carleman set.

We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of the set of complementary discs., Comment: Analysis and PDE, to appear

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., 36 pages