Kirszbraun's extension theorem fails for Almgren's multiple valued functions

TL;DR

This paper demonstrates that Kirszbraun's extension theorem, which allows for the extension of Lipschitz functions, does not hold in the context of Almgren's multiple valued functions, highlighting a fundamental limitation.

Contribution

The paper establishes that an extension theorem similar to Kirszbraun's does not exist for Almgren's multiple valued functions, revealing a key difference from classical Lipschitz theory.

Findings

No Kirszbraun-type extension theorem for Almgren's multiple valued functions

Highlights limitations in extending Lipschitz functions in this setting

Provides insight into the structure of multiple valued functions

Abstract

We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.