Null Curves in ${\mathbb{C}}^3$ and Calabi-Yau Conjectures

Abstract

For any open orientable surface $M$ and convex domain $\Omega \subset {\mathbb{C}}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\to \Omega .$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $\Omega$ in ${\mathbb{C}}^2$ there exist a Riemann surface $N$ homeomorphic to $M$ and a complete proper holomorphic immersion $F:N\to \Omega .$ Furthermore, if $D\subset {\mathbb{R}}^2$ is a convex domain and $\Omega$ is the solid right cylinder $\{x\in {\mathbb{C}}^2|Re(x)\in D\},$ then $F$ can be chosen so that $\mathrm{Re}(F):N\to D$ is proper. There exists a Riemann surface $N$ homeomorphic to $M$ and a complete bounded holomorphic null immersion $F:N\to \mathrm{SL}(2,\mathbb{C}).$ There exists a complete bounded CMC-1 immersion $X:M\to {\mathbb{H}}^3.$ For any convex domain $\Omega \subset {\mathbb{R}}^3$ there exists a complete proper minimal immersion $(X_j{)}_{j=1,2,3}:M\to \Omega$ with vanishing flux. Furthermore, if $D\subset {\mathbb{R}}^2$ is a convex domain and $\Omega =\{(x_j{)}_{j=1,2,3}\in {\mathbb{R}}^3|(x_1,x_2)\in D\},$ then $X$ can be chosen so that $(X_1,X_2):M\to D$ is proper. Any of the above surfaces can be chosen with hyperbolic conformal structure.

20 pages, 4 figures. To appear in Mathematische Annalen

References in corpus (2)

• Density theorems for complete minimal surfaces in R^3
• Existence of proper minimal surfaces of arbitrary topological type

Cited by in corpus (2)

• Compact complete null curves in Complex 3-space
• Calabi-Yau Problem for Legendrian curves in C^3 and applications