Bending-Neutral Deformations of Minimal Surfaces

Abstract

Minimal surfaces are ubiquitous in nature. Here they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that (1) every minimal surface is transformed into a minimal surface by a bending-neutral deformation, (2) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other, and (3) all minimal surfaces have indeed a universal bending content.