Local interpolation for minimal surfaces

TL;DR

This paper demonstrates that for any real analytic curve close to a given one, there exists a minimal surface containing both curves after a translation, expanding understanding of minimal surface interpolation.

Contribution

It introduces a method for constructing minimal surfaces that interpolate between two nearby real analytic curves through local translation techniques.

Findings

Existence of minimal surfaces containing two close real analytic curves.

Construction method for interpolating minimal surfaces.

Extension of classical minimal surface theory to local curve interpolation.

Abstract

Let $a: I\to \mathbb{R}^3 $ be a real analytic curve satisfying some conditions. In this article, we show that for any real analytic curve $l:I\to \mathbb R^3$ close to $a$ (in a sense which is precisely defined in the paper) there exists a translation of $l$, and a minimal surface which contains both $ a $ and the translated $l$.