New Examples of Reflectively Symmetric Minimal Surfaces Bounded by Parallel Straight Lines

Research Report MRR00-011

New Examples of Reflectively Symmetric Minimal Surfaces Bounded by Parallel Straight Lines

Yi Fang and Jenn-Fang Hwang

Abstract: We prove that for each $n\geq 2$, there is a family of properly immersed minimal surfaces $M(n,\phi)$, $0 \leq \phi \leq \pi $, except for $\phi = 0$, they are bounded by parallel straight lines. For $0 \leq \phi < \pi $, $M(n,\phi )$ is invariant under $D_n\times \Bbb Z_2$, where $D_n$ is the general dihedral group, and $\Bbb Z_2$ is generated by a reflection keeping each of the boundary lines invariant. For Each $n \geq 2$, there is a unique constant $\phi _n$, $0 \leq \phi _n < \pi $, such that for $\phi _n < \phi < \pi$, $M(n, \phi )$ is bounded by $2n$ parallel straight lines such that the interior of $M(n, \phi )$ is the union of a minimal graph $G(n, \phi )$ and a reflection of $G(n, \phi )$. In particular, when $\phi _n < \phi < \pi$, $M(n, \phi )$ is embedded. When $n\geq 2$, $G(n, \pi)$ is the Jenkins-Serrin graph over a domain bounded by a regular $2n$-gon; $M(2, 0)$ is a catenoid while for $n\geq 3$, $M(n, 0)$ is the Jorge-Meeks $n$-noid; for $0 < \phi < \pi $, $M(2, \phi )$ is the KMR surface discovered by Karcher, and Meeks and Rosenberg.

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