Related papers: On defining functions for unbounded pseudoconvex d…

We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic.

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties.…

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to…

For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partial\Omega$ can be locally represented as the graph of a…

Let A be a domain of the boundary of a strictly pseudoconvex domain \Omega of C^n and M a smooth, closed, maximally complex submanifold of A. We find a subdomain \widetilde A of \Omega, depending only on \Omega and A, and a complex…

Let $n \geq 3$ and $\Omega$ be a bounded domain in $\mathbb{C}^n$ with a smooth negative plurisubharmonic exhaustion function $\varphi$. As a generalization of Y. Tiba's result, we prove that any holomorphic function on a connected open…

Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily…

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha…

For any pseudoconvex Runge domain $\Omega\subset\mathbb{C}^2$ we prove that every closed discrete subset in $\Omega$ is contained in a properly embedded complex curve in $\Omega$ with any prescribed topology (possibly infinite).

Let $D=\{\rho<0\}$ be a smooth domain of finite type in an almost complex manifold (M,J) of real dimension four. We assume that the defining function $\rho$ is J-plurisubharmonic on a neighborhood of $\overline{D}$. We study the asymptotic…

Let $\Omega \subset \mathbb{R}^n$ be a domain that supports the $p$-Poincar\'e inequality. Given a homeomorphism $\varphi \in L^1_p(\Omega)$, for $p>n$ we show the domain $\varphi(\Omega)$ has finite geodesic diameter. This result has a…

Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is…

If $\Omega$ is a simply connected domain in $\overline{{\mathbb C}}$ then, according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in…

We prove that if a holomorphic self-map $f\colon \Omega\to \Omega$ of a bounded strongly convex domain $\Omega\subset \mathbb C^q$ with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of…

We show that each pseudoconvex domain $\Omega\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hat{\delta}^{-6}}$, where $\hat{\delta}$ is the minimum of the boundary distance and…

In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $\Omega=\omega + i\mathbb{R}^n$ in $\mathbb{C}^n$, where $\omega$ is an open set in $\mathbb{R}^n$, and which is invariant in its…

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\infty)$, $f$ is twice differentiable on…

Let $\Omega\subset\mathbb{C}^n$ be a strictly pseudoconvex Runge domain with $C^2$-smooth defining function, $l\in\mathbb{N},$ $p\in(1,\infty).$ We prove that the holomorphic function $f$ has derivatives of order $l$ in $H^p(\Omega)$ if and…

For each $n\geq2$ we construct an unbounded closed pseudoconcave complete pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety of positive dimension (we call it a \textit{Wermer type set}). We also construct an…

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds…