In the context of the study of the Ohsawa–Takegoshi L² extension theorem, a "qualitative" extension theorem in terms of adjoint ideal sheaves (i.e., extension result without L² estimates but with some form of local L² condition ensured) is proved on compact Kähler manifolds via harmonic theory and residue computations. The arguments are adapted from those in the study of the injectivity theorem by Chan–Choi–Matsumura. The result guarantees, in particular, the existence of extensions of line-bundle-valued holomorphic top forms over the union of any log-canonical centres of the same codimension to top forms over the ambient space under suitable positivity assumptions.

This is a survey article on Schwarz's lemma and some applications. In particular, we study Schwarz–Pick theorems, Lindelöf's inequality, Poincaré metric on the open unit disk, and a δ,ϵ version of Schwarz's lemma, due to Earle. We also discuss an inequality of Harris. We include a brief discussion on quasiconformal mappings and Schwarz's lemma. We conclude with some applications in Teichmüller theory.

We study the extendability problems for proper holomorphic mappings between, possibly different dimensional, domains. After reviewing the known results, we show the following: There exists a proper holomorphic mapping between generalized pseudoellipsoids with different dimensions which can not be extended continuously across the boundary. If we assume the holomorphic extendability, the mappings have normal forms up to automorphisms of the source and the target domains.

Let Δ_𝔹 be the Laplace–Beltrami operator associated with Bergman metric on the unit ball 𝔹 of ℂ^N. We estimate the nonisotropic Hausdorff dimension of a set in ∂𝔹 where a positive Δ_𝔹-superharmonic function u satisfying -Δ_𝔹u(z)≤c(1-|z|)^τu(z)^p in 𝔹 for some p∈(0,N/(N-1)), τ>max{α(p−1),N(p−1)} and c>0 grows faster than a prescribed order α.

In this paper, we first apply a gluing lemma developed in [1] to give an intrinsic and short proof of that the Teichmüller metric coincides with the Kobayashi metric on the Teichmüller space T_α of the C^1+α circle diffeomorphisms for each 0<α<1 and on the Teichmüller space T₁ of the C^1+Zygmund circle diffeomorphisms. Then we introduce characterizations of the elements of T_α and the elements of its tangent space at the base point in terms of conditions on two integral operators studied in [2] and [3] respectively. Some open questions are raised for T₁.

The meromorphic solution of the following type of nonlinear complex differential equation

is considered, where P_n−1(f) is a differential polynomial in f with small function coefficients of degree at most n−1, l is a positive integer. The structure of meromorphic solution f satisfying N(r,f)+N(r,1/f)=S(r,f) is obtained. The result still holds in similar complex differential equations with only one dominant term.

On two subsurfaces of a Riemann surface divided by a p-Weil–Petersson curve γ, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of γ. By requiring the coincidence of boundary values on γ, we establish a correspondence between the harmonic functions in these Banach spaces. We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of p-Besov functions on the unit circle.

The notion of the Yau cycle was introduced by Konno, and the Yau cycles exhibit a strong connection with canonical cycles in certain types of singularities. In this paper, we consider the Brieskorn complete intersection surface singularities, and give the concrete expression of the canonical cycles K_Bi using the Yau i-cycles under certain case.

We study a compact non-orientable Klein surface of genus 6 without boundary such that it contains a disc with the maximum radius determined by its genus. We discuss the group of automorphisms of such a surface. As a result, we find two surfaces that admit an automorphism of order 10, where 10 is the maximum order of automorphisms that compact non-orientable Klein surfaces of genus 6 without boundary can admit. Applying this result, we obtain compact non-orientable Klein surfaces of arbitrary even genus g≥4 admitting an automorphism of maximum order 2(g−1).

Let w be an Erdös-type weight on ℝ and let {p_n} be the orthonormal polynomials with respect to w. We denote by {x_j,n} the zeros of p_n. For an interger ν≥2 and a continuous function f on ℝ, let H_n(ν;f,·) be the ν-th order Hermite–Fejér interpolation polynomial for f based at {x_j,n}, that is, H_n(ν;f,x_j,n)=f(x_j,n) and H_n^(k)(ν;f,x_j,n) = 0 hold for j=1,2,…,n and k=1,2,…,ν-1. We discuss a uniform norm of H_n(ν;f,·)-f on ℝ and give quantitative interpolation estimates.

In 2016, Beberok obtained an explicit formula for the Bergman kernel of a specific Reinhardt domain in ℂ³, defined by |ζ₁|²<(1−|z|²)^q,|ζ₂|²<(1−|z|²)^r where q,r>0. We derive explicit formulas for the Bergman kernels of a wider class of domains, expressed in terms of polylogarithm functions. This class includes Beberok's domain as a special case.