Uroš Kuzman
We prove that every $J$-holomorphic variational vector field can be realized as derivation $\frac{d}{dt}_{|t=0}f_t$ where $(f_t)$ is a one parametric family of $J$-holomorphic discs. Furthermore, we discuss properness of an extremal $J$-holomorphic disc in a bounded pseudoconvex domain.
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We prove that the category of generalized holomorphically contractible families possesses maximal and minimal objects. Moreover, we present basic properties of these extremal families.
Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.
In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball, classical Cartan domains, symmetrised bidisc and tetrablock. In particular we show that weak $3$-extremal maps in the symmetrised bidisc are rational thus giving the (partial) answer to a problem posed in a recent paper by J. Agler, Z. Lykova and N. J. Young.
We prove that a function, which is defined on a union of lines $\mathbb{C} E$ through the origin in $\mathbb{C}^n$ with direction vectors in $E\subset \mathbb{C}^n$ and is holomorphic of fixed finite order and finite type along each line, extends to an entire holomorphic function on $\mathbb{C}^n$ of the same order and finite type, provided that $E$ has positive homogeneous capacity in the sense of Siciak and all directional derivatives along the lines satisfy a necessary compatibility condition at the origin. We are able to estimate the indicator function of the extension in terms of Siciak's…
We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be call…
We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $Γ$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_ν, ν\geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show…
We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay Theorem on holomorphic discs and our recent joint-work with Pflug on cross theorems in dimension 1. It also relies on our new technique of conformal mappings and a generalization of Siciak's relative extremal function. Our approach illustrates the unified character: ``From local informations to global extensions". Moreover, it avoids systematically the use of the classical method of doubly orthogonal bases of Bergman type.
Extremal rate of convergence in continuous dynamics
本文刻画上半平面全纯自映射半群以最慢速率收敛至Denjoy--Wolff点的条件，重点解决抛物情形，并利用Herglotz表示与Koenigs函数共形性提供多种等价描述。
Answers to three problems in Dinh-Sibony's list on holomorphic foliation theory
本文解决 Dinh-Sibony 列表中三个问题：证明饱和全纯叶状正闭电流有限 Poincaré 质量，构造具有稠密双曲叶和弥散极值电流的射影曲面，并给出双曲叶 Nevanlinna 特征的对数下界。
The main purpose of this article is to present a generalization of Forelli's theorem for the functions holomorphic along a general pencil of holomorphic discs. This generalizes the main result of \cite{JKS13} and the original Forelli's theorem, and furthermore, answers one of the problems posed in \cite{Chirka06}.
We generalize Lempert's and Poletsky's works on the description of extremal discs for the Kobayashi metric to a higher order setting.
We provide a local approximation result of non-holomorphic discs with small d-bar by pseudoholomorphic ones. As an application, we provide a certain gluing construction.
We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function $q$ on an open connected subset $X$ in $\C^n$. This function is also known as the weighted Green function with logaritmic pole at infinity and weighted global extremal function.
We prove that for any complex manifold X, the set of all holomorphic maps from the unit disc to X whose images are everywhere dense in X forms a dense subset in the space of all holomorphic maps from the disc to X. We show by an example that this need not hold for maps from the disc to complex spaces with singularities.
We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.
Let $(ϕ_t)$ be a semigroup of holomorphic self-maps of~$\mathbb D$. In this note, we use an abstract approach to define the Königs function of $(ϕ_t)$ and "holomorphic models" and show how to deduce the existence and properties of the infinitesimal generator of $(ϕ_t)$ from this construction.
We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.
A computer-assisted proof of Krzyż's conjecture for the sixth coefficient
本研究针对Krzyż猜想的第六个系数，通过结合Agler与McCarthy的归约方法、Lei与Zhang的熵定理，采用严格的计算机辅助分支定界法，排除了极值原子数少于六个的情况，完成了该猜想的计算机辅助证明，同时该方法也独立验证了已知情形的全次数性质。
We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.
We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and Carathéodory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex domains are also established.
We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.
We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.
Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for any path p:[0,1)-->U such that |p(t)|-->1 as t-->1, the path t--> f(p(t)) from [0,1) to B has infinite length.
We show that there is no complete proper holomorphic map from the open disc U in C to the bidisc UxU which extends continuously to the closed disc.
We show how to extend some holomorphic bundles with fifer C^2 and base an open set in C, to bundles on the Riemann Sphere, by an extremely simple technique. In particular, it applies to examples of non-Stein bundles constructed by Skoda and Demailly. It gives an example on C, with polynomial transition automorphisms.
Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.
We construct knotted proper holomorphic embeddings of the unit disc in C^2.
We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.
We prove analogs of Thom's transversality theorem and Whitney's theorem on immersions for pseudo-holomorphic discs. We also prove that pseudo-holomorphic discs form a manifold.
We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc, the total speed, the orthogonal speed and the tangential speed and show how they are related and what can be inferred from those.
We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori.
We introduce Hausdorff operators over the unit disc and give conditions for boundedness of such operator in Bloch, Bergman, and Hardy spaces on the disc. Identity approximation by Hausdorff operators is also considered.
We prove that the set of leaves of a holomorphic lamination of codimension one that are non-transversal to a germ of a holomorphic map is discrete.
In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is everywhere dense in the sphere $b\mathbb{B}^2$.
We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup $(ϕ_t)$ in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to the hyperbolic metric of its Koenigs domain $Ω$ and the latter one deals with Euclidean properties of $Ω$.
We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of Hölderian norms of such discs.
The existence criterion of holomorphic discs for higher $A_\infty$ operations via minimal discs
本文通过极小盘研究更高$A_\infty$操作的存在性，证明极小盘边界在Lagrangian子流形上相交时，其部分Maslov指数满足特定条件则存在对应的全纯盘。
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