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## 1. Mappings between balls

1. ### mapping problem 1

#### Problem 1.1.

[B. Lamel] Let $n \ge 2$.
• 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.

• 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?
Open (test)
• #### Problem 1.2.

[F. Meylan] 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 $\mathbb{C}^{n}$?
1. Remark. [org.aimpl.user:rshroff@math.ucsd.edu] the codimension $1$ case is known (Chiappari, ’91)
• #### Problem 1.3.

[X. Huang] 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?
• #### Problem 1.4.

[X. Huang] 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?
• #### Problem 1.5.

[J. D’Angelo] 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 $f_{1} = p$.

Cite this as: AimPL: Complexity of CR Mappings, available at http://aimpl.org/crmap.