Compact complete null curves in Complex 3-space

Abstract

We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset \mathcal{M}$ and a continuous map $X:\bar{M}\to {\mathbb{C}}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain no relatively compact components in $\mathcal{M},$ $X{|}_M$ is a complete null holomorphic curve, $X{|}_{\bar{M}-M}:\bar{M}-M\to {\mathbb{C}}^3$ is an embedding and the Hausdorff dimension of $X(\bar{M}-M)$ is $1.$ Moreover, for any $\epsilon >0$ and compact null holomorphic curve $Y:N\to {\mathbb{C}}^3$ with non-empty boundary $Y(\partial N),$ there exist Riemann surfaces $M$ and $\mathcal{M}$ homeomorphic to $N^{\circ }$ and a map $X:\bar{M}\to {\mathbb{C}}^3$ in the above conditions such that ${\delta }^H(Y(\partial N),X(\bar{M}-M))<\epsilon ,$ where ${\delta }^H(\cdot ,\cdot )$ means Hausdorff distance in ${\mathbb{C}}^3.$

17 pages, 3 figures

References in corpus (1)

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

Cited by in corpus (1)

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