Mathematics

We investigate decay near boundary of the volume of sublevel sets in Cegrell classes of m− subharmonic function on bounded domains in C n . On the reverse direction, some sufficient conditions for membership in certain Cegrell's classes, in terms of the decay of the sublevel sets, are also discussed.

If f is a non-Archimedean analytic curve in a projective variety X embedded in P N and if D 1 ,…, D q are hypersurfaces of P N in general position with X, then we prove the defect relation: ∑ j=1 q δ(f, D j )≤dimX.

Abstract deformations of the CR structure of a compact strictly pseudoconvex hypersurface M in C 2 are encoded by complex functions on M . In sharp contrast with the higher dimensional case, the natural integrability condition for 3 -dimensional CR structures is vacuous, and generic deformations of a compact strictly pseudoconvex hypersurface M⊆ C 2 are not embeddable even in C N for any N . A fundamental (and difficult) problem is to characterize when a complex function on M⊆ C 2 gives rise to an actual deformation of M inside C 2 . In this paper we study the embeddability of families of deformations of a given embedded CR 3 -manifold, and the structure of the space of embeddable CR structures on S 3 . We show that the space of embeddable deformations of the standard CR 3 -sphere is a Frechet submanifold of C ∞ ( S 3 ,C) near the origin. We establish a modified version of the Cheng-Lee slice theorem in which we are able to characterize precisely the embeddable deformations in the slice (in terms of spherical harmonics). We also introduce a canonical family of embeddable deformations and corresponding embeddings starting with any infinitesimally embeddable deformation of the unit sphere in C 2 .

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball B m ( R 0 ) in C m ( 0< R 0 ≤+∞ ). In this article, we will show that if three meromorphic mappings f 1 , f 2 , f 3 of M into P n (C) (n≥2) satisfying the condition ( C ρ ) and sharing q (q>C+ρK) hyperplanes in general position regardless of multiplicity with certain positive constants K and C<2n (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for k (2≤k≤n+1) meromorphic mappings sharing hyperplanes is also given. Our result generalize the previous result in the case where the mappings from C m into P n (C) .

Let $H^\infty(\mathbb D\times\N)$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk D⊂C . We show that the dense stable rank of $H^\infty(\mathbb D\times\N)$ is one and using this fact prove some nonlinear Runge-type approximation theorems for $H^\infty(\mathbb D\times\N)$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for algebra H ∞ (D) .

We compare the notion of relative non-pluripolar products and that of density currents on compact Kahler manifolds.

This paper is a survey of main developments in Oka theory since the publication of my book "Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)", Second Edition, Springer, Cham, 2017. The paper is self-contained to the extent possible and is accessible also to readers who are new to the field. It will be updated periodically and available at this https URL.

We propose the concept of Diederich--Fornæss and Steinness indices on compact pseudoconvex CR manifolds of hypersurface type in terms of the D'Angelo 1-form. When the CR manifold bounds a domain in a complex manifold, under certain additional non-degeneracy condition, those indices are shown to coincide with the original Diederich--Fornæss and Steinness indices of the domain, and CR invariance of the original indices follows.

In this paper, we study the boundedness and compactness of the differences of two weighted composition operators acting from α -Bloch space to β -Bloch space on the open unit disk. This study has a relationship to the topological structure of weighted composition from α -Bloch space to β -Bloch space.

In this paper, some characterizations for the compact difference of composition operators on Bergman spaces A p ω with doubling weight are given, which extend Moorhouse's characterization for the difference of composition operators on the weighted Bergman space A 2 α .