Chapter 6 - Examples of Area-Minimizing Surfaces

Abstract

This chapter discusses examples of area-minimizing surfaces. The minimal surface equation gives the necessary condition that under smooth variations in the surface, the rate of change of the area is 0. This condition turns out to be equivalent to the vanishing of the mean curvature. A smoothly immersed surface which is locally the graph of a solution to the minimal surface equation (or, equivalently, which has mean curvature 0) is called a minimal surface. It presents figures of some famous minimal surfaces. The minimal surface theorem guarantees that small pieces of minimal surfaces are area mini- mizing, but larger pieces may not be. For example, the portion of Enneper's surface is not area minimizing. There are two area-minimizing surfaces with the same boundary and some systems of curves in R3 bound infinitely many minimal surfaces. On a disc (or any other convex domain), there is a solution of the minimal surface equation with any given continuous boundary values. The proof is omitted because the minimal surface equation is not linear, this fact is not at all obvious, and it fails if the domain is not convex.