On the genus of triply periodic minimal surfaces
Let \(M\) be a triply periodic minimal surface in Euclidean space, and let \(\Lambda\) be the lattice generated by its three periods. Then \(M\) projects to a minimal surface in the flat \(3\)-torus \({\mathbb R}^3/\Lambda\), and conversely a non-flat minimal surface in \({\mathbb R}^3/\Lambda\) lifts to a triply periodic minimal surface in \({\mathbb R}^3\). In this paper, for fixed \(\Lambda\) and fixed \(g\geq 3\), the author constructs minimal surfaces of genus \(g\) in \({\mathbb R}^3/\Lambda\) with arbitrarily large area and, in case \(g\geq 4\), having the property that the identity is the only translation of \({\mathbb R}^3/\Lambda\) that leaves his surfaces fixed. This work proves a conjecture of \textit{W. Meeks} [Indiana Univ. Math. J. 39, No.3, 877--936 (1990; Zbl 0721.53057)] and answers a question of [\textit{H. Karcher}, Geometry and topology III, Proc. Workshop, Leeds/UK 1990, 174--191 (1991; Zbl 0737.53007)]. The author describes the idea behind his construction as follows: Choose a plane in \(3\)-space so that its projection in \({\mathbb R}^3/\Lambda\) has large area. In \({\mathbb R}^3/\Lambda\), the plane will wrap around many times creating many parallel sheets close together. Then take another parallel plane and open small catenoidal necks between the two planes to control the genus of the surface. Of course, the placement of the necks cannot be arbitrary and substantial effort is required to carry out the plan.
scientific article; zbMATH DE number 5294962