On the localization principle for the automorphisms of pseudoellipsoids

Abstract:

We show that Alexander’s extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $f$ of a pseudoellipsoid $\mathcal {E}^n_{(p_1, \dots , p_{k})}\overset {\text {def}}{=} \{ z \in \mathbb {C}^n : \sum _{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + \dots + |z_n|^{2 p_{k}} < 1\}$, provided that $f$ is defined on a region $\mathcal {U} \subset \mathcal {E}^n_{(p)}$ such that: i) $\partial \mathcal {U} \cap \partial \mathcal {E}^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\mathcal {U}\cap \{ z_i = 0 \} \neq \emptyset$ for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.