The Calabi-Yau conjectures for embedded surfaces

By T. Colding, W. Minicozzi. 2004

175 citations recorded by the source corpus.

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.”

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cf839673a0fbcab7e2daa938e4588cd1d5edf5ff

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In Applied Mathematics

1. W. H. Meeks
2. Joaquín Pérez
3. A. Ros
4. H. Rosenberg
5. W. Minicozzi
6. T. Colding
7. Reto Muller
8. Leili Shahriyari

In Geometry and Topology

1. W. H. Meeks
2. Joaquín Pérez
3. A. Ros
4. H. Rosenberg
5. W. Minicozzi
6. T. Colding
7. Reto Muller
8. J. Lott

Citations in the corpus

• Stable constant mean curvature surfaces (2008, background): “In the minimal case, this result was found independently by Colding and Minicozzi [7] and it is motivated by an earlier almost identical result in [38] and by still earlier work of Gulliver and Lawson [24].”
• MAXIMUM PRINCIPLES AT INFINITY (2008, background): “Results in [ 5 ] and [17] imply that every complete, embedded minimal surface of finite genus and one end is properly embedded and intersects some plane transversely in a single component, and so, is recurrent for Brownian motion; in this case, our theorem in this section gives an independent proof ”
• The classical theory of minimal surfaces (2011, methodology): “2 is a refinement of the results and techniques used by Colding and Minicozzi [37] to prove the following deep result (see Section 9 where we deduce Theorem 1.”
• The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus (2005, methodology): “Proof. The first claim (C.5) follows precisely from the proof of lemma 2.23 in [CM9] that is given in appendix B in [ CM9 ].53”