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 ≥ 2, we prove that there is a unique constant φn, 0 ≤ φn < π, and there exists a family of properly embedded minimal surfaces M(n, φ), φn < φ < π. Each M(n, φ) is bounded by 2n parallel straight lines such that the interior of M(n, φ) is the union of a minimal graph G(n, φ) and its reflection. Each M (n, φ) is invariant under Dn × ℤ2, where Dn is the general dihedral group, and ℤ2 is generated by a reflection keeping each of the boundary lines invariant. The graph G(n, φ) is over an non-convex bounded domain with a Jenkins-Serrin type capillary boundary values. Moreover, for each n ≥ 2, M (n, φ) can be put in a bigger family of immersed minimal surfaces, 0 ≤ φ ≤ π. For φ < 0 < π, M(n, φ) has the same symmetric property and is bounded by parallel straight lines. When n ≥ 2, G(n, π) is the Jenkins-Serrin graph over a domain bounded by a regular 2n-gon; M(2, 0) is a catenoid while for n ≥ 3, M (n, φ) is the Jorge-Meeks n-noid; for 0 < φ < π, M(2, φ) is the KMR surface discovered by Karcher, and Meeks and Rosenberg. Thus in the moduli space of properly immersed minimal surfaces, M(n, φ), n ≥ 2, 0 ≤ φ π, is a connected path connecting the catenoid or Jorge-Meeks n-noid to the Jenkins-Serrin graph. © Birkhäuser Verlag, 2002.