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The Calabi-Yau conjectures for embedded surfaces

Tobias H. Colding·William P. Minicozzi II
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Abstract

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.

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Tobias H. Colding, William P. Minicozzi II. 2006-11-07. The Calabi-Yau conjectures for embedded surfaces. https://arxiv.org/abs/math/0404197

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