We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂⁿ and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

Let each point of a homogeneous Poisson process in ℝ^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution μ on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution μ. Leaving aside degenerate cases, we prove that for any μ there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on μ for the absence and presence of infinite components.

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set K defined on the boundary of the unit ball.

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂⁿ to complex transversal wedges with edges containing N. We show that given any v∈ℂⁿ∖(T_pN+iT_pN), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.

We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least k and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where n≥3, k≥1 and 0<|x|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple f of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with f when f defines a complete intersection.

The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces Jⁿ of submanifolds of M. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on Jⁿ persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

Let M be a smooth hypersurface of constant signature in CPⁿ, n≥3. We prove the regularity for $\bar{∂}_b$ on M in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in CPⁿ, n≥3, whose Levi form has at least two zero-eigenvalues.