Unique continuation for area minimizing currents

TL;DR

This paper proves a unique continuation principle for area minimizing currents, showing that infinite order contact with a minimal surface implies local coincidence, advancing understanding of geometric measure theory.

Contribution

It establishes a new unique continuation result for area minimizing currents, linking infinite contact order to local equivalence with minimal surfaces.

Findings

Infinite order contact implies local coincidence of currents and surfaces.

Provides a rigorous proof of unique continuation in geometric measure theory.

Enhances understanding of the structure of area minimizing currents.

Abstract

The main goal of this work is to prove an instance of the unique continuation principle for area minimizing integral currents. More precisely, consider an $m$-dimensional area minimizing integral current and an $m$-dimensional minimal surface, both contained in $\mathbb{R}^{n+m}$ with $n\geq 1$. We show that if, in an integral sense, the current has infinite order of contact with the minimal surface at a point, then the current and the minimal surface coincide in a neighborhood of that point.