相关论文: Continuity of capacity of a holomorphic motion

We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the…

We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the…

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property…

In this paper we examinate some phenomena arising when a holomorphic germ is analytically continued.

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U…

We present a comprehensive survey on removability of compact plane sets with respect to various classes of holomorphic functions. We also discuss some applications and several open questions, some of which are new.

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay…

We study functions which are the pointwise limit of a sequence of holomorphic functions. In one complex variable this is a classical topic, though we offer some new points of view and new results. Some novel results for solutions of…

Consider a continuous one parameter family of circles in complex plane that contains two circles lying in the exterior of one another. Under mild assumptions on the family, we prove that if a continuous function on the union of the above…

Real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball. This was proved in a previous preprint by an argument…

Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function…

Flexible mechanical metamaterials possess repeating structural motifs that imbue them with novel, exciting properties including programmability, anomalous elastic moduli and nonlinear and robust response. We address such structures via…

We describe holomorphic functions and fractional powers of Ces\'{a}ro operators in $L^2(\mathbb{R})$, $L^2(\mathbb{R}_+)$, and $L^2[0,1]$. Logarithms of Ces\'{a}ro operators are introduced as well and their spectral properties are studied.…

We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain…

In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.

The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity…

A general kind of models with hierarchically constrained dynamics is shown to exhibit logarithmic anomalous relaxation, similarly to a variety of complex strongly interacting materials. The logarithmic behavior describes most of the decay…

We give a brief review of holomorphic motions and its relation with quasiconformal mapping theory. Furthermore, we apply the holomorphic motions to give new proofs of famous Konig's Theorem and Bottcher's Theorem in classical complex…

We study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential. We establish local well-posedness and propose two possible weak formulations that exist…

The new results concerning the continuity of holomorphically contractible systems treated as set functions with respect to non-monotonic sequences of sets are given. In particular, continuity properties of Kobayashi and Carath\'eodory…