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

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.