Related papers: Uniqueness in Rough Almost Complex Structures and …

Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.

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…

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic…

n this article we consider functions meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions which also contains some known results. We include few open problems for…

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact…

We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. Combining results of Clemente [3], we show that applying h-principle…

Consider complex semisimple Lie algebras of a given dimension specified by their structure constants. We describe a finite collection of rational functions in the structure constants that form a complete set of invariants: two sets of…

We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, H\"older regularity, and fractional differentiability. Our first main result shows that for…

While small deformations of K\"ahler manifolds are K\"ahler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently…

We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence K\"ahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant…

We study the existence and stability of holomorphic foliations in dimension greater than 4 under perturbations of the underlying almost complex structure. An example is given to show that, unlike in dimension 4, J-holomorphic foliations are…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus…

This paper is about the integrability of complex vector fields in dimension three in a neighborhood of a singular point. More precisely, we study the existence of holomorphic first integrals for isolated singularities of holomorphic vector…

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…

We show that there is an almost complex structure on a differential calculus on finite points coming from a bidirected finite graph without multiple edges or loops. We concentrate on a polygon as a concrete case. In particular, a…

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of…

We consider a class of weighted harmonic functions in the open upper half-plane known as $\alpha$-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the…

In this paper, we calculate the dimension of the $J$-anti-invariant cohomology subgroup $H_J^-$ on $\mathbb{T}^4$. Inspired by the concrete example, $\mathbb{T}^4$, we get that: On a closed symplectic $4$-dimensional manifold $(M, \omega)$,…

We give a condition for an almost constant-type manifold to be a constant-type manifold, and holomorphic and $R$-invariant submanifolds of almost Hermitian manifolds are studied. Generalizations of some results in [5] are given.

In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…