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1. Mappings between balls

    1. mapping problem 1


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      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)


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          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. [Ravi Shroff] the codimension 1 case is known (Chiappari, ’91)


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                  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?


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                      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?


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                          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.