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 exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

• For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

• There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

• 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 → Ω 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 → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.