Related papers: Critical sets of proper holomorphic mappings
Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.
We prove that the graph of a continuous function $f$, defined on a domain of ${\mathbb C}^n$, is pluripolar if and only if $f$ is holomorphic.
Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then…
Paradigms of bilinear maps f between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of f with Keller C^n_c-maps (on suitable…
We investigate regularity of CR-mappings between real-analytic infinite type hypersurfaces in $\mathbb C^2$. We show that, under the Fuchsian type condition, all (respectively formal or smooth) CR-diffeomorphisms between them are…
Let $F : H^q \to H^q$ be a $C^k$-map between Sobolev spaces, either on $\mathbb R^d$ or on a compact manifold. We show that equivariance of $F$ under the diffeomorphism group allows to trade regularity of $F$ as a nonlinear map for…
A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev,…
Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving…
Given a pair of curves C_1 and C_2 on a hyperbolic surface F, when does there exist a pseudo-Anosov map sending one to another? More generally, one may ask the same question for C_i to be sets of disjoint simple closed curves. We will give…
We investigate the dynamical behaviour of a holomorphic map on a $f-$invariant subset $\mathcal{C}$ of $U,$ where $f:U \to \mathbb{C}^k.$ We study two cases: when $U$ is an open, connected and polynomially convex subset of $\mathbb{C}^k$…
We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty,…
This paper discusses a more general contractive condition for a class of extended cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same…
We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…
The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it…
We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert…
On a reasonable class of domains in $\CC^n$, we characterize those holomorphic functions which continue analytically past the boundary. Then we give some applications of this result to holomorphic mappings. In addition, some new results…
We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call…
Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein…
It is shown that if $f$ and $g$ are any two analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of $f$ to that of $g$ has regularity $C^{1+\alpha}$ at the critical…
We focus on a sequence of functions $\{f_n\}$, defined on a compact manifold with boundary $S$, converging in the $C^k$ metric to a limit $f$. A common assumption implicitly made in the empirical sciences is that when such functions…