Examples of holomorphic functions vanishing to infinite order at the boundary

TL;DR

This paper constructs examples of holomorphic functions that vanish infinitely at boundary points, illustrating complexities in boundary regularity and implications for minimal surfaces and Q-valued functions.

Contribution

It provides explicit examples of boundary-vanishing holomorphic functions and discusses their implications for regularity, branching, and unique continuation in related mathematical contexts.

Findings

Holomorphic functions can vanish to infinite order at boundary points.

Such functions influence the regularity and branching behavior in minimal surface theory.

Implications for unique continuation and boundary regularity in Q-valued functions.

Abstract

We present examples of holomorphic functions that vanish to in- finite order at points at the boundary of their domain of definition. They give rise to examples of Dirichlet minimizing Q-valued functions indicating that "higher"-regularity boundary results are difficult. Furthermore we dis- cuss some implication to branching and vanishing phenomena in the context of minimal surfaces, Q-valued functions and unique continuation.