Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.1726

Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps

Xianghong Gong·Jean-Pierre Rosay
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $Ω$ be an open subset in $\mathbf C$ and let $E$ be a closed polar subset of $Ω$. Let $u$ be a continuous map from $Ω$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $Ω\setminus E$ then it is J-holomorphic on $Ω$.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xianghong Gong, Jean-Pierre Rosay. 2007-08-13. Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps. https://arxiv.org/abs/0708.1726

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous spaces of entire functions

Homogeneous spaces are de Branges' Hilbert spaces of entire functions with the property that certain weighted rescaling transforms induce isometries of the space into itself. A classical example of a homogeneous space is the Paley-Wiener space of entire functions with exponential type at most a being square integrable on the real axis. Other examples occur in the theory of the Bessel equation. Being homogeneous is a strong property, and one can describe all homogeneous spaces, their structure Hamiltonians, and the measures associated with chains of such spaces, explicitly in terms of powers, logarithms, and confluent hypergeometric functions. The theory of homogeneous spaces was in large parts settled by L.de Branges in the early 1960's. However, in his work some connections and explicit formulae are not given, some results are stated without a proof, and last but not least a mistake occurs which seemingly remained unnoticed up to the day. In this paper we give a detailed account on homogeneous spaces. We provide explicit proofs for all formulae and relations between the mentioned objects, and correct the mentioned mistake.

math.CV↗
arXiv2026

Logarithmic coefficients for exponential classes of starlike and convex functions

In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $φ(z)=e^{αz},\qquad 0<α\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{αz} \quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{αz}$. These classes provide a natural exponential analogue of several classical subclasses arising in geometric function theory. We obtain sharp coefficient estimates, logarithmic coefficient inequalities and sharp bounds for the associated Hankel and upper bounds for Toeplitz determinants. In particular, explicit estimates are derived for $$ |H_{2,1}(F_f/2)|, \quad |T_{2,1}(F_f/2)|, $$ for functions belonging to the introduced exponential subclasses of starlike and convex functions. Our results extend and unify several earlier works on exponential subclasses and highlight connections with logarithmic coefficients and determinant functionals.

math.CV↗
arXiv2026

The curvature estimation of the complete Kähler-Einstein metrics on the disk bundles

In this paper, we study some geometric properties of the disk bundles over the complete Kähler manifolds. When the base space is the complete Kähler-Einstein manifold, we compute the holomorphic sectional curvature, Riemannian sectional curvature of the complete Kähler-Einstein metric on the disk bundle. Moreover, we present a parametric criteria for whether the manifold admits negative pinched property. When the base space is a bounded pseudoconvex domain equipped with its complete Kähler metric, the corresponding ball bundle in the $k$-direct sum of line bundles is a bounded pseudoconvex Hartogs domain with fiber dimension $k$ . If the base domain is simply-connected and $k\geq2$, we prove that the Bergman metric on such a Hartogs domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to a unit ball.

math.CV↗