Minimal surfaces with micro-oscillations

TL;DR

This paper demonstrates the construction of minimal graphs in higher-dimensional space with controlled small oscillations, allowing for prescribed geometric intersections with a hyperplane, expanding understanding of minimal surface configurations.

Contribution

It introduces a method to construct minimal graphs with small oscillations that can be tailored to have specific geometric intersections, a novel approach in minimal surface theory.

Findings

Existence of minimal graphs with prescribed intersection geometry.

Construction of almost flat minimal graphs with controlled oscillations.

Extension of minimal surface examples with customizable boundary behavior.

Abstract

We show that there are minimal graphs in R^{n+1} whose intersection with the portion of the horizontal hyperplane contained in the unit ball has any prescribed geometry, up to a small deformation. The proof hinges on the construction of minimal graphs that are almost flat but have small oscillations whose geometry we can control.