Surveys in Differential Geometry

Volume 9 (2004)

Conformal properties in classical minimal surface theory

Pages: 275 – 335

DOI: https://dx.doi.org/10.4310/SDG.2004.v9.n1.a8

Authors

W. H. Meeks (Mathematics Department, University of Massachusetts, Amherst, MA 01003, U.S.A.)

J. Pérez (Universidad de Granada, Departamento de Geometria y Topologia, Fuentenueva s/n, 18071 Granada, Spain.)

Abstract

This is a survey of recent developments in the classical theory of minimal surfaces in R³ with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover the maximum principle at infinity for properly immersed minimal surfaces in R³ and some new results on harmonic functions as they relate to the classical theory. We define and demonstrate the usefulness of universal superharmonic functions. We present the compactness and regularity theory of Colding and Minicozzi for limits of sequences of simply connected minimal surfaces and its application by Meeks and Rosenberg in their proof of the uniqueness of the helicoid. Finally, we discuss some recent deep results on the topology and on the index of the stability operator of properly embedded minimal surfaces and give an application of the classical Shiffman Jacobi function to the classification of minimal surfaces of genus zero.

Published 1 January 2004