  In this paper we will prove the Calabi-Yau conjectures for embedded surfaces.
In fact, we will prove considerably more.
  The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work
has been done on them over the past four decades. In particular, examples of
Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed
versions were false; we will show here that for embedded surfaces, i.e.,
injective immersions, they are in fact true.
