Thomas Wolff Memorial Lecture in Mathematics

Abstract: The first example of a bounded, complete minimal surface was given in the 1970's (PJ). Since then several new types of such surfaces have been shown to exist. I will discuss recent research (PJ) showing that there exists a new type of example, namely a complete, bounded Legendrian disk in C^3. This means there are two bounded holomorphic functions (F, G) on the unit disk such that (F(z), G(z),z) is a complete, bounded minimal surface, with the property that (setting H'(z) = F(z)G'(z)) H is also a bounded holomorphic function. The construction uses a modification of Uchiyama's method of decomposing BMO functions into u + H(v), where u and v are bounded functions. I will also discuss some open problems in this area.