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Defects in Triply Periodic Minimal Surfaces

三周期极小曲面中的缺陷

基本信息

批准号：
398759432
负责人：
Dr. Hao Chen
金额：
--
依托单位：
Institut für Numerische und Angewandte Mathematik (NAM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/398759432?language=en
关键词：
Defects Triply Periodic Minimal Surfaces

项目摘要

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.

三周期极小曲面（TPMSs）是一种奇妙的几何物体，它将肥皂泡的美丽与局部最小化区域和具有重复图案的晶体相结合。它们在生物学和材料科学中被广泛观察到。建议的项目旨在将晶体缺陷（即周期图案的不完美）引入TPMS。该项目于2018年6月展开。它是由材料科学的观察激发的。该项目的主要成就是使用Traizet的节点开放技术严格构建平面缺陷。此外，通过有意打破对称性，令人惊讶的新的属3的TPMS在近20年来首次明确构建。到目前为止，这些结果大大扩展了我们对TPMS的知识，拓宽了我在数学和物理方面的研究范围。鉴于到目前为止的成就，扩展将更多地关注严格的数学构造。特别是，我计划通过胶合Scherk塔来构造线缺陷，并研究胶合螺旋面的可能性。这些项目再次受到生物系统中观察到的结构的激励。这些结果不仅有助于我们进一步理解TPMS，而且有助于我们理解一般极小曲面的模空间。