Defects in Triply Periodic Minimal Surfaces

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Triply periodic minimal surfaces (TPMSs) are fantastic geometric objects that combine the beauty of soap bubbles with locally minimized areas and crystals with repeating patterns. They are widely observed in biology and material science. The proposed project aims to introduce crystallographic defects, i.e. imperfections of the periodic patterns, into TPMSs.The project started in June 2018. It was motivated by observations in material sciences. The major achievement of the project is the rigorous constructions of planar defects using Traizet's node-opening technique. Moreover, by intentionally breaking symmetries, surprising new TPMSs of genus three were explicitly constructed for the first time in almost 20 years. The results so far significantly expanded our knowledge of TPMSs and widened my scope of research both in mathematics and physics.I propose to renew the project for another year. In view of the achievement so far, the extension will focus more on rigorous mathematical constructions. In particular, I plan to construct line defects by gluing Scherk towers and investigate the possibility of gluing helicoids. These projects are again motivated by structures observed in biological systems. The results are expected to further improve our understanding not only on TPMSs but also on the moduli space of minimal surfaces in general.