c°2006, Sociedade Brasileira de Matem¶atica RECENT PROGRESSES IN THE CALABI-YAU PROBLEM FOR MINIMAL SURFACES

In the last forty years, interest of many geometers and analysts has concentrated on the global theory of complete minimal surfaces. Be-cause there were no su±ciently complicated examples for exact investi-gation, this new development proceeded only slowly. However, last few years have seen an important progress on many long-standing problems in global theory of complete minimal surfaces in R3. One of these has been the Calabi-Yau problem, which dates back to the 1960s. Calabi asked whether or not it is possible for a complete minimal surface in R3 to be contained in the ball B = fx 2 R3 j kxk < 1g: Much work has been done on it over the past four decades. The most important result in this line was obtained by N. Nadirashvili in [24] where he constructed a complete minimal surface in B. After Nadirashvili's work, Calabi-Yau problem continues generating literature. Questions related with embeddedness and properness of com-pleted bounded minimal surfaces has been particularly interesting. In this survey we pretend to review the most important advances in this area.