The Calabi–Yau conjectures for embedded surfaces

Abstract

In this talk I will discuss the proof of the Calabi-Yau conjectures for embedded surfaces. This is joint work with Bill Minicozzi, [CM9]. 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. Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (see also S.S. Chern, page 212 of [Ch]): Conjecture 1. “Prove that a complete minimal hypersurface in R must be unbounded.” Calabi continued: “It is known that there are no compact minimal submanifolds of R (or of any simply connected complete Riemannian manifold with sectional curvature ≤ 0). A more ambitious conjecture is”: Conjecture 2. “A complete minimal hypersurface in R has an unbounded projection in every (n− 2)–dimensional flat subspace.”