Two New Embedded Triply Periodic Minimal Surfaces of Genus 4

Abstract: We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are orthogonal in the second case. The second family has as one limit the Costa surface, showing that this limit can occur for triply periodic minimal surfaces. For the existence proof we solve the 1-dimensional period problem through a combination of an asymptotic analysis of the period integrals and geometric methods. v2: corrected embeddedness proof. Minor further improvements.