The book is an overview of the research on minimal surfaces of the last few years written by several renowned researchers. The methods used are taken from differential geometry, complex analysis, calculus of variations and the theory of partial differential equations. Readers will be graduate students and researchers in mathematics.

Klappentext

Few people outside of mathematics are aware of the varieties of mathemat­ ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas­ ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function.

Inhalt

I. Complete Embedded Minimal Surfaces of Finite Total Curvature.- II. Nevanlinna Theory and Minimal Surfaces.- III. Boundary Value Problems for Minimal Surfaces.- IV. The Minimal Surface Equation.- Author Index.