Calabi-Yau Problem for Legendrian curves in C^3 and applications
  We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3. 
Francisco Martin, Masaaki Umehara, Kotaro Yamada
math.DG