Dihedralization of Minimal Surfaces in $\mathbb{R}^3$

TL;DR

This paper investigates the limits of minimal surfaces with dihedral symmetry as the wedge angle approaches zero, revealing simpler limit surfaces and enabling new existence proofs and discoveries of minimal surfaces.

Contribution

It introduces a novel approach using limits and the implicit function theorem to prove existence and find new minimal surfaces with small dihedral angles.

Findings

Limits of minimal surfaces are simpler as the dihedral angle approaches zero.

The method provides new proofs of existence for minimal surfaces with small dihedral angles.

New minimal surfaces are discovered through this limiting process.

Abstract

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes with varying angle. We will study the limit of such surfaces when the angle converges to 0. In many cases, these limits are simpler than the original surface, and can be used in conjunction with the implicit function theorem to give new existence proofs of the original surfaces with small dihedral angle. This approach has led to the discovery of new minimal surfaces as well.