Defects in Triply Periodic Minimal Surfaces

• Applicant: Dr. Hao Chen
• Subject Area: Mathematics ; Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 398759432

• Final Report Year: 2021

Final Report Abstract

Triply periodic minimal surfaces (TPMSs) are fantastic geometric objects that combine the beauty of bubble films and crystals: They minimize local area and repeat themselves throughout the space. The study of TPMSs traces back to Schwarz in the 1860s, but new examples are discovered in a very slow pace. Since the 1970s, physicists and chemists became aware of ubiquitous applications of TPMSs, and began to contribute in the topic. They provided many important new examples, some are exotic as they do not belong to the 5-parameter family of Meeks that contains most examples known at the time. The most famous exotic example would be the Gyroid. Mathematicians struggled to provide rigorous treatment for the exotic examples. The current project is again motivated by natural sciences. In 2011, Han et al. observed a TPMS structure in laboratory that is not perfect, but looks like two perfect TPMSs glued along a mirror. It was not clear that this defected structure is still a TPMS. The PI helped Han to numerically reproduce the structure, but was not satisfied by numerical approach. The initial goal of the project is to develop a rigorous mathematical treatment of defected TPMSs. For this purpose, the PI enlists the help of Traizet, who constructed minimal surfaces that look like horizontal planes connected by catenoid necks. In the limit, the planes are very close to each other, and the catenoid necks are very tiny. The idea of construction is to push the planes a little bit away from each other. The PI and Traizet improved the technique to allow non-periodic arrangement of the necks. It turns out that the arrangement can be so crazy that crystallographers would see them as “disorders”. The observed defected TPMSs are then rigorously justified as a special case of disorder. To return the favor, the PI worked closely with natural scientists to provide mathematical insights and explanations for the structures they observed in laboratories. In crystallography, knowledge of defects often leads to deeper insights into perfect crystals. It is then no surprise that knowledge of defects would lead to the discovery of new perfect TPMSs. In collaboration with Matthias Weber, the PI constructed two new families of TPMSs. These are the first new examples since decades. Moreover, the PI provided an existence proof for deformations of the Gyroid. They were known to physicists in the 1990s, but never rigorously proved before. These progresses are all motivated or inspired by defected TPMSs. The new examples are all “exotic” examples outside Meeks’ family, but they all belong to families that intersect the Meeks family. The intersections are known as bifurcations: As one deforms the lattice, the TPMS usually deforms in a unique way; but at a bifurcation, the TPMS has several choices to deform. The new examples are the first explicit examples of interesting bifurcations. Saddle towers are minimal surfaces that look like intersecting planes from afar. The project was extended for another year for the objective to glue saddle towers into minimal surfaces. In collaboration with Traizet, the PI revealed a subtle interaction between saddle towers that was never perceived before. As a consequence, many new TPMSs are discovered. The project can be considered as the most significant progress in the theory of TPMSs since decades. In the future, the PI plans to improve his techniques to construct more new examples of TPMSs, making progress towards an ultimate classification for TPMSs with simplest topology.

Publications

• Minimal twin surfaces. Exp. Math., 28(4):404–419, 2019
  Hao Chen
  (See online at https://doi.org/10.1080/10586458.2017.1413455)
• Crystal twinning of bicontinuous cubic structures. IUCrJ, 7(2):228–237, 2020
  Lu Han, Nobuhisa Fujita, Hao Chen, Chenyu Jin, Osamu Terasaki, and Shunai Che
  (See online at https://doi.org/10.1107/S2052252519017287)
• Existence of the tetragonal and rhombohedral deformation families of the gyroid. Indiana Univ. Math. J., 70(4):1543–1576, 2021
  Hao Chen
  (See online at https://doi.org/10.1512/iumj.2021.70.8505)
• Stacking disorder in periodic minimal surfaces. SIAM J. Math. Anal., 53(1):855–887, 2021
  Hao Chen and Martin Traizet
  (See online at https://doi.org/10.1137/20M1312137)