Minimal surfaces with the area growth of two planes; the case of infinite symmetry

TL;DR

This paper classifies certain symmetric minimal surfaces in Euclidean 3-space, showing that under specific area growth constraints, they must be planes, catenoids, or Scherk surfaces, with implications for desingularization of intersecting planes.

Contribution

It establishes a classification of symmetric minimal surfaces with specific area growth, identifying them as planes, catenoids, or Scherk surfaces, and characterizes the unique desingularization of intersecting planes.

Findings

Connected minimal surfaces with infinite symmetry are classified as planes, catenoids, or Scherk surfaces.

The only periodic minimal desingularization of intersecting planes is Scherk's surface.

Surfaces with area growth less than 2πR^2 are constrained to these classical forms.

Abstract

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.