CyberSec Research

We show that when $m>n$, the space of $m\times n$-matrix-valued rational inner functions in the disk is path connected.

In this paper, we establish a quantitative correspondence between power quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings. In particular, we prove that the exponents in the power quasi-symmetric mappings coincide with the coefficients in the rough quasi-isometric mappings. This shows that the obtained correspondence is both sharp and consistent. In this way, we generalize the corresponding result by Bj\"orn, Bj\"orn, Gill, and Shanmugalingam (J. Reine Angew. Math., 2017) from the setting of rooted trees to that of hyperbolic fillings.

Given an $n$-dimensional compact K\"ahler manifold, we continue our study of $m$-positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree $(1,\,1)$ by relaxing the standard positivity hypotheses to their $m$-counterparts after we have proved a Lamari-type duality lemma in bidegree $(m,\,m)$. Independently, we propose a Monge-Amp\`ere-type non-linear pde whose distinctive feature is that its solutions, if any, are forms of positive degree rather than functions. We prove a form of uniqueness for the solutions and, under the assumption that a solution exists, we give a geometric application involving the $m$-bigness notion introduced in the first part.

This article surveys the impact of Eremenko and Lyubich's paper ''Examples of entire functions with pathological dynamics'', published in 1987 in the Journal of the LMS. Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics.

The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of complex manifolds with a good supply of global meromorphic functions, in particular, it includes all Stein manifolds and projective manifolds. It is also shown that there exist noncompact complex manifolds, known as long $\mathbb C^2$, that are {\calligra M }-manifolds but do not contain any nonconstant holomorphic functions.

We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and BMOA does not necessarily lie in VMOA, and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.

In this article, we give completely new examples of embedded complex manifolds the germ of neighborhood of which is holomorphically equivalent to a germ of neighborhood of the zero section in its normal bundle. The first set of examples is composed of connected abelian complex Lie groups, embedded in some complex manifold $M$. These are non compact manifolds in general. We also give some conditions ensuring the existence a holomorphic foliation having the embedded manifold as leaf. The second set of examples are $n$-dimensional Hopf manifolds, embedded as hypersurfaces.

Let $X$ be a smooth open manifold of even dimension, $T$ be a topological space, and $\mathscr{J}=\{J_t\}_{t\in T}$ be a continuous family of smooth integrable Stein structures on $X$. Under suitable additional assumptions on $T$ and $\mathscr{J}$, we prove an Oka principle for continuous families of maps from the family of Stein manifolds $(X,J_t)$, $t\in T$, to any Oka manifold, showing that every family of continuous maps is homotopic to a family of $J_t$-holomorphic maps depending continuously on $t$. We also prove the Oka-Weil theorem for sections of $\mathscr{J}$-holomorphic vector bundles on $Z=T\times X$ and the Oka principle for isomorphism classes of such bundles. The assumption on the family $\mathscr{J}$ is that the $J_t$-convex hulls of any compact set in $X$ are upper semicontinuous with respect to $t\in T$; such a family is said to be tame. For suitable parameter spaces $T$, we characterise tameness by the existence of a continuous family $\rho_t:X\to \mathbb{R}_+=[0,+\infty)$, $t\in T$, of strongly $J_t$-plurisubharmonic exhaustion functions on $X$. Every family of complex structures on an open orientable surface is tame.We give an example of a nontame smooth family of Stein structures $J_t$ on $\R^{2n}$ $(t\in \mathbb{R},\ n>1)$ such that $(\mathbb{R}^{2n},J_t)$ is biholomorphic to $\mathbb{C}^n$ for every $t\in\mathbb{R}$. We show that the Oka principle fails on any nontame family.

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Amp\`ere-type equation and the associated Dirichlet problem.

In this paper we show that Baker domains of transcendental skew products can either bulge or not, depending on the higher order terms. This is in contrast to polynomial skew products where all Fatou components with bounded orbits of an invariant attracting fiber do bulge.

In K\"ahler geometry, the Donaldson-Fujiki moment map picture interprets the scalar curvature of a K\"ahler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,\omega)$, let $\mathcal{G}=\mathrm{Aut}(P)$. We construct for each $k\in\mathbb{N}$ a $\mathcal{G}$-equivariant determinant line bundle $\lambda^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $\lambda^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed two-forms $\Omega_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $\mu_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $\mu_j$ arises from the asymptotic expansion of the variation of the log of the Quillen metric with respect to K\"ahler potentials, keeping the complex structure fixed. This provides a natural generalization of the Donaldson-Fujiki moment map interpretation of scalar curvature. Moreover, we show that $\mu_j$ coincide with the $Z$-critical equations introduced by Dervan-Hallam, and we state a generalization of Fujiki's fiber integral formula.

We consider the canonical ensemble of a system of point particles on the sphere interacting via a logarithmic pair potential. In this setting, we study the associated Gibbs measure and partition function, and we derive explicit formulas relating the critical temperature, at which the partition function diverges, to a certain discrete optimization problem. We further show that the asymptotic behavior of both the partition function and the Gibbs measure near the critical temperature is governed by the same optimization problem. Our approach relies on the Fulton--MacPherson compactification of configuration spaces and analytic continuation of complex powers. To illustrate the results, we apply them to well-studied systems, including the two-component plasma and the Onsager model of turbulence. In particular, for the two-component plasma with general charges, we describe the formation of dipoles close to the critical temperature, which we determine explicitly.

We analyze the behavior of multipliers of a degenerating sequence of complex rational maps. We show either most periodic points have uniformly bounded multipliers, or most of them have exploding multipliers at a common scale. We further explore the set of scales induced by the growth of multipliers. Using Ahbyankar's theorem, we prove that there can be at most 2d-2 such non-trivial multiplier scales.

In this paper, we introduce the notions of the $k$-th Milnor number and the $k$-th Tjurina number for a germ of holomorphic foliation on the complex plane with an isolated singularity at the origin. We develop a detailed study of these invariants, establishing explicit formulas and relating them to other indices associated with holomorphic foliations. In particular, we obtain an explicit expression for the $k$-th Milnor number of a foliation and, as a consequence, a formula for the $k$-th Milnor number of a holomorphic function. We analyze their topological behavior, proving that the $k$-th Milnor number of a holomorphic function is a topological invariant, whereas the $k$-th Tjurina number is not. In dimension two, we provide a positive answer to a conjecture posed by Hussain, Liu, Yau, and Zuo concerning a sharp lower bound for the $k$-th Tjurina number of a weighted homogeneous polynomial. We also present a counterexample to another conjecture of Hussain, Yau, and Zuo regarding the ratio between these invariants. Moreover, we establish a fundamental relation linking the $k$-th Tjurina numbers of a foliation and of an invariant curve via the G\'omez-Mont--Seade--Verjovsky index, and we extend Teissier's Lemma to the setting of $k$-th polar intersection numbers. In addition, we derive an upper bound for the $k$-th Milnor number of a foliation in terms of its $k$-th Tjurina number along balanced divisors of separatrices. Finally, for non-dicritical quasi-homogeneous foliations, we obtain a closed formula for their $k$-th Milnor and Tjurina numbers.

In this paper a survey is given of application of a method based on Grunsky coefficients for obtaining different estimates (some sharp) for the general class of univalent functions where no analytical characterisation exists. More precisely, estimates are given for the modulus of the third and the fourth logarithmic coefficients, for the modulus of the second and the third Hankel determinant for the general class of univalent functions, and for the modulus of some coefficients of the inverse function, and some coefficient differences.

A commuting pair of Hilbert space operators having the closed symmetrized bidisc \[ \Gamma=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\} \] as a spectral set is called a \textit{$\Gamma$-contraction}. A $\Gamma$-contraction $(S,P)$ is called \textit{$\Gamma$-distinguished} if $(S,P)$ is annihilated by a polynomial $q \in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ defines a distinguished variety in the symmetrized bidisc $\mathbb G$. There is Schaffer-type minimal $\Gamma$-isometric dilation of a $\Gamma$-contraction $(S,P)$ in the literature. In this article, we study when such a minimal $\Gamma$-isometric dilation is $\Gamma$-distinguished provided that $(S,P)$ is a $\Gamma$-distinguished $\Gamma$-contraction. We show that a pure $\Gamma$-isometry $(T,V)$ with defect space $\dim \mathcal D_{V^*}< \infty$, is $\Gamma$-distinguished if and only if the fundamental operator of $(T^*,V^*)$ has numerical radius less than $1$. Further, it is proved that a $\Gamma$-contraction acting on a finite-dimensional Hilbert space dilates to a $\Gamma$-distinguished $\Gamma$-isometry if its fundamental operator has numerical radius less than $1$. We also provide sufficient conditions for a pure $\Gamma$-contraction to be $\Gamma$-distinguished. Wold decomposition splits an isometry into two orthogonal parts of which one is a unitary and the other is a completely non-unitary contraction. In this direction, we find a few decomposition results for the $\Gamma$-distinguished $\Gamma$-unitaries and $\Gamma$-distinguished pure $\Gamma$-isometries.

We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature and a corresponding variational characterization. All simply connected discrete minimal surfaces of this type can be constructed from circle patterns via a discrete Weierstrass representation formula. This representation links the space of discrete minimal surfaces to the deformation space of circle patterns, and thereby to classical Teichm\"uller theory. We also discuss variants of discrete minimal surfaces obtained by modifying the definition of mean curvature, restricting the variational criterion, or replacing circle pattern data with discrete conformal equivalence, Koebe-type circle packings, or quadrilateral meshes with factorized cross ratios. We conclude with open questions on discrete minimal surfaces.