Related papers: A Landau's theorem in several complex variables
The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic…
The Landau-Pekar equations describe the dynamics of a strongly coupled polaron. Here we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this…
In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy…
In this paper, we employ the theory of normal families in several complex variables to obtain some uniqueness theorems for entire functions. These results extend the related works of Li and Yi [11], and Lu et al. [18] to the setting of…
We show that any two holomorhpic maps, not both of which are constant, from a generalized Hopf manifold to its base must have a coincidence. We prove a similar result for holomorphic maps from a generalized Calabi-Eckmann manifold.
In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent…
We develop a theory of variable elliptic structures on planar domains, in which the imaginary unit $i(x,y)$ is a moving generator of a rank-two real algebra bundle defined by a smoothly varying quadratic relation. Differentiating this…
We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally…
In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $\Omega$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that…
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a…
Let $B^n$ be the unit ball in $\mathbb C^n$ and let the points $a_1,...,a_{n+1} \in B^n $ are affinely independent. If $f \in C(\partial B^n)$ and for any complex line $L$, containing at least one of the points $a_j$, the restriction $f|_{L…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…
We show that for every continuous function there exists an absolutely continuous homeomorphism of the circle such that the Fourier series of the composition converges uniformly. This resolves a problem set by N. N. Luzin.
We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an…
A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real…
We consider the boundary dynamics of iterated function systems of holomorphic self-maps of the unit disc. Our main result provides a sufficient condition which guarantees that the dynamical behaviour of a left iterated function system in…
Classic complex analysis is built on structural function $K=1$ only associated with Cauchy-Riemann equations, subsequently various generalizations of Cauchy-Riemann equations start to break this situation. The goal of this article is to…
An existence theorem for stationary discs of strongly pseudoconvex domains in almost complex manifolds is proved. More precisely, it is shown that, for all points of a suitable neighborhood of the boundary and for any vector belonging to…
In this paper we present a proof of Hartogs' extension theorem, following T. Sobieszek's paper from 2003. Hartogs' theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is "a…
Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain…