Abstract

A minimal surface in euclidean space is a surface which is locally of least area (that is, any perturbation on a small region will increase the area). These surfaces have captured the imagination of geometers and analysts from Riemann and Weierstrauss to the present day, where they have evolved to become an important tool in modern geometric analysis.

This lecture will give an introduction to the basic geometry of minimal surfaces in R³. Several of the classical theorems will be presented, and a large number of interesting examples (such as triply-periodic minimal surfaces) will be examined.

In the second part of the lecture I will discuss minimal surfaces in a more general context. Examples will include compact minimal surfaces of arbitrary genus in the euclidean 3-sphere S³, complex algebraic curves in complex projective space, and certain “calibrated” subvarieties. The problem of finding area-minimizing cycles in a given homology class will be discussed. I will then try to indicate the role played by minimal surfaces in modern geometry, topology and physics.