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1986•Transactions of the American Mathematical SocietyRequires access

Deformations of Complete Minimal Surfaces

Harold Rosenberg

Open publisher page 10 citations

Abstract

A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and ${R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.

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What this paper is about

A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and ${R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.

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Available abstract

A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and ${R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.

Key concepts: Helicoid, Minimal surface, Mathematics, Surface (topology), Mean curvature, Geometry, Curvature, Combinatorics

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