New examples of reflectively symmetric minimal surfaces bounded by parallel straight lines

Abstract.

In this paper, we give examples of proper embedded Jenkins-Serrin type minimal surfaces with a reflective symmetry. That is, for each$n \geq 2$, we prove that there is a unique constant$\phi _n, 0 \leq \phi _n < \pi$, and there exists a family of properly embedded minimal surfaces $M(n, \phi), \phi_n < \phi < \pi$. Each $M(n, \phi)$ 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 its reflection. Each $M(n,\phi)$is invariant under $D_{n}\times\mathbb{Z}_{2}$, where $D_{n}$ is the general dihedral group, and$\mathbb{Z}_{2}$ is generated by a reflection keeping each of the boundary lines invariant. The graph $G(n,\phi)$ is over an non-convex bounded domain with a Jenkins-Serrin type capillary boundary values.¶¶Moreover, for each $n\geq 2, M(n, \phi)$ can be put in a bigger family of immersed minimal surfaces, $0\leq \phi \leq \pi$. For $0 < \phi < \pi, M(n, \phi)$has the same symmetric property and is bounded by parallel straight lines.¶¶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.¶¶Thus in the moduli space of properly immersed minimal surfaces,$M(n, \phi), n \geq 2, 0 \leq \phi \leq \pi$, is a connected path connecting the catenoid or Jorge-Meeks n-noid to the Jenkins-Serrin graph.