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

Abstract

We prove that a connected properly immersed minimal surface in E 3 {\mathbb E}^3 with infinite symmetry group and area growth constant less than 3 π 3\pi is a plane, a catenoid, or a Scherk singly-periodic minimal surface. As a consequence, the Scherk minimal surfaces are the only connected periodic minimal desingularizations of the intersections of two planes.