A Burns-Krantz type theorem for domains with corners

TL;DR

This paper establishes local boundary uniqueness theorems for holomorphic maps near corners of domains, extending classical results to more general boundary conditions and higher codimension submanifolds.

Contribution

It introduces new boundary uniqueness results for maps defined only on one side and extends these results to submanifolds of higher codimension.

Findings

Proves boundary uniqueness for maps with weaker asymptotic conditions.

Extends boundary results from domain boundaries to higher codimension submanifolds.

Provides new insights into boundary behavior of holomorphic maps near corners.

Abstract

The goal of this paper is twofold. First, to give purely local boundary uniqueness results for maps defined only on one side as germs at a boundary point and hence not necessarily sending any domain to itself and also under the weaker assumption that $f(z)=z+o(|z-p|^3)$ holds only for $z$ in a proper cone in $D$ with vertex $p$. Such results have no analogues in one complex variable in contrast to the situation when a domain is preserved. And second, to extend the above results from boundaries of domains to submanifolds of higher codimension.