In this note, we give an example of a domain whose d-balanced squeezing function is nonplurisubharmonic.

We prove a version of the Fatou theorem for bounded functions with a bounded $\overline \partial _J$ part of the differential on wedge-type domains in an almost complex manifold.

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.

We prove that a compact subset of full measure on a generic submanifold of an almost complex manifold is not a pluripolar set. Several related results on boundary behavior of plurisubharmonic functions are established. Our approach is based on gluing a family of complex discs to a generic manifold along a boundary arc and could admit further applications.

Suppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$

$p_{j}\geqslant 1$

$N=1+n_{1}+\cdots +n_{k}$

$w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$

$\unicode[STIX]{x1D706}_{D}$

$D$

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$

$\unicode[STIX]{x1D6FA}_{N}(D)$

$D$

$\mathbb{C}^{2}$

$\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$

$\unicode[STIX]{x1D6FC}$

$f$

$\unicode[STIX]{x1D6FC}$

We establish a Schwarz lemma for $V$-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ${{\mathbb{C}}^{n}}$.

Let ${{B}_{p1,p2}}\,=\,\left\{ z\,\in \,{{\mathbb{C}}^{n}}\,:\,{{\left| {{z}_{1}} \right|}^{{{p}_{1}}}}\,+\,{{\left| {{z}_{2}} \right|}^{{{p}_{2}}}}\,+\,\cdots \,+\,{{\left| {{z}_{n}} \right|}^{{{p}_{2}}}}\,<\,1 \right\}$ be an egg domain in ${{\mathbb{C}}^{n}}$. In this paper, we first characterize the Kobayashi metric on ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge 1,\,{{p}_{2}}\,>\,1 \right)$ and then establish a new type of classical boundary Schwarz lemma at ${{z}_{0}}\in \partial {{B}_{{{p}_{1}},{{p}_{2}}}}$ for holomorphic self-mappings of ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge \,1,\,{{p}_{2}}\,>\,1 \right)$), where ${{z}_{0}}={{\left( {{e}^{i\theta }},\,0,\ldots ,0 \right)}^{\prime }}$ and $\theta \,\in \,\mathbb{R}$.

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ${{\mathbb{C}}^{n}}$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\,\,\|z\|)$ on ${{B}^{n}}\backslash \{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$.

Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

In this paper, the characterization of domains in ℂn by their noncompact automorphism groups are given.

We give an explicit upper bound on the number of non-constant holomorphic maps from a quasi-projective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi.

We propose a procedure to construct new smooth CR-manifolds whose local stability groups, equipped with their natural topologies, are subgroups of certain (finite-dimensional) Lie groups but not Lie groups themselves.

The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let ${{V}_{n}}={{\mathbb{S}}^{2n+1}}\times {{\mathbb{S}}^{2n+1}}$ denote a Calabi-Eckmann manifold. If $f,g:\,{{V}_{n}}\to {{\mathbb{P}}^{n}}$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)\,=\,g(x)$ for some $x\in {{V}_{n}}$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\,\subset \,G$ is a maximal parabolic subgroup, where one of the maps is dominant.

Let $B$ be the unit ball of ${{\mathbb{C}}^{n}}$ with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from $B$ into ${{\mathbb{C}}^{n}}$ of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of $B$ which arises in the study of the Loewner equation, namely the set ${{S}^{0}}\left( B \right)$ of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of ${{S}^{0}}\left( B \right)$ obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of ${{S}^{0}}\left( B \right)$ as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in $S(B)$ which can be imbedded in Loewner chains, but which do not have parametric representation.

Given an almost arbitrary holomorphic map we study the structure of the associated residue integral and its Mellin transform, and the relation between these two objects. More precisely, we relate the limit behaviour of the residue integral to the polar structure of the Mellin transform. We consider also ideals connected to nonisolated singularities.

In this paper, we study the boundary version of the classical Cartan theorem. We show that for some weakly pseudoconvex domains, when a holomorphic self-mapping has a sufficiently high order of contact (which depends only on the geometric properties of the domains) with the identical map at some boundary point, then it must coincide with the identity.