On the localization principle for the automorphisms of pseudoellipsoids

TL;DR

This paper extends Alexander's theorem on automorphisms from the unit ball to pseudoellipsoids, establishing conditions under which local automorphisms can be globally extended, with optimal hypotheses demonstrated by counterexamples.

Contribution

It generalizes the extendibility theorem for automorphisms to pseudoellipsoids and identifies optimal conditions for such extensions.

Findings

Extension of Alexander's theorem to pseudoellipsoids

Conditions for local automorphisms to extend globally

Counterexamples showing optimality of hypotheses

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 $\E^n_{(p_1, ..., p_{k})} \= \{z \in \C^n : \sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + ... + |z_n|^{2 p_{k}} < 1 \}$, provided that $f$ is defined on a region $\U \subset \E^n_{(p)}$ such that: i) $\partial \U \cap \partial \E^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\U \cap \{z_i = 0 \} \neq \emptyset$ for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.