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Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings…

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be…

We show that, in toric manifolds, one can characterize the sign of the Ricci curvature in terms of the convexity of the volume functional. More generally we discuss relationships between (i) Ricci curvature and volume, (ii) totally real and…

Given a compact polarized manifold $(X,L)$, we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on $X$. We prove that in the Fano setting, the new invariants can effectively…

We prove the existence and study the geometry of Bishop discs near an elliptic point of a real n-dimensional submanifold of an almost complex n-dimensional manifold.

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a…

We study Wiener-type covering lemmas, Hardy-Littlewood-type maximal functions, and convergence theorems on metric spacs. Later we specialize down to a result for the Poisson integral. We show that, in a suitably general setting, these three…

We study functionals on the space of almost complex structures on a compact $\mathbb{C}$-manifold, whose variational properties could be used to tackle Yau's Challenge.

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of…

Given a $\mathbb Z_2$-harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction…

Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the…

We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on…

We give a characterization of Inoue surfaces in terms of automorphic pluriharmonic functions on a cyclic covering. Together with results of Chiose and Toma, this completes the classification of compact complex surfaces of Kaehler rank one.

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $\SU n$, $\SO n$ and $\Sp n$. We work in a geometric setting which connects our…

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular…

We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous…

In this paper we prove the basic facts for pluricomplex Green functions on manifolds. The main goal is to establish properties of complex manifolds that make them analogous to relatively compact or hyperconvex domains in Stein manifolds.…

Let $(V,\omega)$ be a compact K\"ahler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $\omega$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an…

This paper can be considered as an extension to our paper [On symplectically harmonic forms on six-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001), n 1, 89-109]. Also, it contains a brief survey of recent results on symplectically…

We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.