Calabi-Yau Problem for Legendrian curves in C^3 and applications
Francisco Martin, Masaaki Umehara, Kotaro Yamada
TL;DR
This paper constructs a complete bounded Legendrian immersion in complex three-space and applies it to generate new examples of bounded flat fronts in hyperbolic, de Sitter, and affine spaces, advancing geometric understanding.
Contribution
It introduces the first known complete bounded Legendrian immersion in C^3 and demonstrates its applications to various geometric front types.
Findings
First example of a complete bounded Legendrian immersion in C^3
Construction of weakly complete bounded flat fronts in hyperbolic and de Sitter spaces
Construction of a weakly complete bounded improper affine front in R^3
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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Analytic and geometric function theory