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\begin{document}
\title{Complexity of CR Mappings}
\author{Edited by Ravi Shroff}

\urlstub{crmap}

\maketitle

problem list 


\section{Mappings between balls}





\begin{problemblock}
\name{mapping problem 1}


\begin{problem}[1.1]  Let $n \ge 2$.
\begin{itemize}
\item Given a proper rational map $f: \mathbb{B}^{n} \rightarrow \mathbb{B}^{N}$, find the smallest number $k(n, N)$ such that $f$ is determined by its $k$-jet at the origin.

\item Let $\Omega, \Omega'$ in $\mathbb{C}^{n}$ and $\mathbb{C}^{N}$ respectively be strongly pseudoconvex domains.  If $f: \Omega \rightarrow \Omega'$ is a proper holomorphic map which extends smoothly to $\partial \Omega$, does there exist a $k$ such that $f$ is determined by its $k$-jet at a point?
\end{itemize}
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.2] Assume $M \subseteq \mathbb{C}^{n}$ is a real-analytic generic submanifold, $f = \frac{p}{q}:\mathbb{C}^{n} \rightarrow \mathbb{C}^{N}$ is the germ of a meromorphic map at $z \in M$, and $f$ is holomorphic in a one-sided wedge $W$ attached to $M$ near $z$.  Assume that $\Vert f(w)\Vert^{2}$ increases to $1$ as $w \rightarrow M$  in $W$ .  Does $f$ extend to a full neighborhood of $z$ in $\mathbb{C}^{n}$?

 \end{problem}




\begin{remark}the codimension $1$ case is known (Chiappari, '91)\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.3] Does there exist a universal constant $t$ such that a proper holomorphic map between $\mathbb{B}^{n}$ and $\mathbb{B}^{N}$ (with $1 < n < N$) which is $\mathcal{C}^{t}$ up to the boundary is actually a rational map? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.4] Assume $\Gamma \subseteq \textrm{Aut}(\mathbb{B}^{N})$ is a discrete subgroup and $f: \mathbb{B}^{n} \rightarrow \mathbb{B}^{N}$ is a proper holomorphic embedding such that $\Gamma(f(\mathbb{B}^{n})) \subseteq f(\mathbb{B}^{n})$ and $f(\mathbb{B}^{n})/\Gamma$ is compact.  Is $f$ linear? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.5] Let $f:\mathbb{B}^{n} \rightarrow \mathbb{B}^{N}$ be a proper rational map.  Is $f$ homotopic to a polynomial map $p$ of the same degree?  That is, $\forall t \in [0, 1]$, each $f_{t}:\mathbb{B}^{n} \rightarrow \mathbb{B}^{N}$ is proper with $f_{0} = f$ and $f_{1} = p$. \end{problem}




\end{problemblock}





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\end{document}
