Calabi-Yau Problem for Legendrian curves in C^3 and applications

Abstract

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.

After the submission, the authors found that the proof of the main theorem contains a serious error and the main theorem turns to be an open problem. ArXiv:1205.4779 is a new paper showing the existence of a weakly complete flat surface (with singularities) in hyperbolic 3-space such that the images of the hyperbolic Gauss maps are bounded, which plays a role of a recovery of the present paper

References in corpus (3)

• Applications of a completeness lemma in minimal surface theory to various classes of surfaces
• Null Curves in ${\mathbb{C}}^3$ and Calabi-Yau Conjectures
• Compact complete null curves in Complex 3-space

Cited by in corpus (1)

• A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space