arXiv 1106.06842011-06-27math.DG

复三维空间中的紧完备零曲线

Compact complete null curves in Complex 3-space

Antonio Alarcon, Francisco J. Lopez

备注：: 17 pages, 3 figures

摘要：

证明了对任意有限拓扑的开可定向曲面$S$，存在一个黎曼曲面$\mathcal{M},$，一个相对紧的区域$M\subset\mathcal{M}$和一个连续映射$X:\bar{M}\to\mathbb{C}^3$使得：$\mathcal{M}$和$M$同胚于$S,$ $\mathcal{M}-M$和$\mathcal{M}-\bar{M}$，$X|_M$不含$\mathcal{M},$中的相对紧分量，$X|_{\bar{M}-M}:\bar{M}-M\to\mathbb{C}^3$是嵌入，并且$X(\bar{M}-M)$的Hausdorff维数是$1.$。对于任意$ε>0$和紧零全纯曲线$Y:N\to\mathbb{C}^3$，空边界$Y(\partial N),$存在与$N^\circ$同胚的黎曼表面$M$和$\mathcal{M}$以及在上述条件下的映射$X:\bar{M}\to\mathbb{C}^3$，使得$δ^H(Y(\partial N),X(\bar{M}-M))<ε,$，其中$δ^H(\cdot,\cdot)$表示$\mathbb{C}^3.$中的Hausdorff距离

英文摘要：

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 $ε>0$ 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 $δ^H(Y(\partial N),X(\bar{M}-M))<ε,$ where $δ^H(\cdot,\cdot)$ means Hausdorff distance in $\mathbb{C}^3.$

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摘要：

英文摘要：

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