Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1102.1333

Estimates for solutions of the $\partial$-equation and application to the characterization of the zero varieties of the functions of the nevanlinna class for lineally convex domains of finite type

Philippe Charpentier·Yves Dupain·Modi Mounkaila
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the late ten years, the resolution of the equation $\bar\partial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As the method used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly a solution of the $\bar\partial$-equation but only a representation formula which allows us to end the resolution of the equation using Kohn's $L^2$ theory. As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.

Explore related subjects

math.CV

Keep this discovery

BibTeXRIS

Philippe Charpentier, Yves Dupain, Modi Mounkaila. 2011-02-07. Estimates for solutions of the $\partial$-equation and application to the characterization of the zero varieties of the functions of the nevanlinna class for lineally convex domains of finite type. https://doi.org/10.1007/s12220-013-9398-5

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

KEEP EXPLORING

Related papers

arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗
arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.

math.CV↗
arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗