Minimal Surfaces Bounded by Parallel Lines

Research Report MRR99-040

Minimal Surfaces Bounded by Parallel Lines

Yi Fang, Jenn-Fang Hwang

Abstract: Let $L = \bigcup _{1\leq j\leq n, 1\leq k \leq 2} L_{j,k}$ be $2n$ parallel straight lines. We prove that if the symmetry group of $L$ contains a subgroup isomorphic to the dihedral group $D_n$, $n \geq 3$, then there is a family of properly embedded, non-flat minimal surfaces bounded by $L$. In the limit case, we get two doubly periodic, properly immersed minimal surfaces of genus 3 and 5, with 8 ends. The self-intersections of these surfaces are parallel straight lines. We also give Enneper-Weierstrass representations of connected, non-flat, properly immersed minimal surfaces that have a reflectional symmetry and $2n \geq 4$ flat band ends and are bounded by $2n$ parallel straight lines. The representations have $5n - 4$ real parameters satisfying a system of at most $5n - 4$ restraining equations. We estimate such surfaces via their boundaries, thus in some cases we can prove that the system of restraining equations has no solutions.

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