### A boundary rigidity problem for holomorphic mappings.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

In this survey we give geometric interpretations of some standard results on boundary behaviour of holomorphic self-maps in the unit disc of ℂ and generalize them to holomorphic self-maps of some particular domains of ℂⁿ.

In this paper, the definition of the derivative of meromorphic functions is extended to holomorphic maps from a plane domain into the complex projective space. We then use it to study the normality criteria for families of holomorphic maps. The results obtained generalize and improve Schwick's theorem for normal families.

We give several extensions to unbounded domains of the following classical theorem of H. Cartan: A biholomorphism between two bounded complete circular domains of Cn which fixes the origin is a linear map. In our paper, pseudo-convexity plays a main role. Some precise study is done for the case of dimension two and the case where one of the domains is Cn.

Let $X$ be a complex Banach space. Recall that $X$ admits afinite-dimensional Schauder decompositionif there exists a sequence ${\left\{{X}_{n}\right\}}_{n=1}^{\infty}$ of finite-dimensional subspaces of $X,$ such that every $x\in X$ has a unique representation of the form $x={\sum}_{n=1}^{\infty}{x}_{n},$ with ${x}_{n}\in {X}_{n}$ for every $n.$ The finite-dimensional Schauder decomposition is said to beunconditionalif, for every $x\in X,$ the series $x={\sum}_{n=1}^{\infty}{x}_{n},$ which represents $x,$ converges unconditionally, that is, ${\sum}_{n=1}^{\infty}{x}_{\pi \left(n\right)}$ converges for every permutation $\pi $ of the integers. For short, we say that $X$ admits an unconditional F.D.D.We...

Let X be a nonsingular complex algebraic curve and let Y be a nonsingular rational complex algebraic surface. Given a compact subset K of X, every holomorphic map from a neighborhood of K in X into Y can be approximated by rational maps from X into Y having no poles in K. If Y is a nonsingular projective complex surface with the first Betti number nonzero, then such an approximation is impossible.

We show that the local automorphism group of a minimal real-analytic CR manifold $M$ is a finite dimensional Lie group if (and only if) $M$ is holomorphically nondegenerate by constructing a jet parametrization.

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form$$\phantom{\rule{0.166667em}{0ex}}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in ${\mathbb{R}}^{3}$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

For a strongly pseudoconvex domain $D\subset {\u2102}^{n+1}$ defined by a real polynomial of degree ${k}_{0}$, we prove that the Lie group $\mathrm{Aut}\left(D\right)$ can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle $Y$ of $\partial D$, and that the sum of its Betti numbers is bounded by a certain constant ${C}_{n,{k}_{0}}$ depending only on $n$ and ${k}_{0}$. In case $D$ is simply connected, we further give an explicit but quite rough bound in terms of the dimension and the degree of the defining polynomial. Our approach is to adapt the Cartan-Chern-Moser...

We obtain an extension of Jack-Miller-Mocanu’s Lemma for holomorphic mappings defined in some Reinhardt domains in ${\u2102}^{n}$. Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain ${B}_{2p}$ with p ≥ 1.