Calabi-Yau Problem for Legendrian curves in C^3 and applications
Differential Geometry
2012-05-24 v2
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.
Keywords
Cite
@article{arxiv.1107.0106,
title = {Calabi-Yau Problem for Legendrian curves in C^3 and applications},
author = {Francisco Martin and Masaaki Umehara and Kotaro Yamada},
journal= {arXiv preprint arXiv:1107.0106},
year = {2012}
}
Comments
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