Today’s Research in Minimal Surfaces and Their Applications to Our World

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Image from Quanta Magazine

Current Research

Research in minimal surfaces has continued throughout the years, but recently, Karen Uhlenbeck, a U.S. mathematician, was named to be one of the most influential as her work led to remarkable advances in the last 40 years. In March 2019, Uhlenbeck became the first woman to be awarded the prestigious Abel Prize, which she received for her “pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics” (Saplakoglu).

 In her early work, Uhlenbeck and mathematician Jonathan Sacks studied minimal surfaces and soap films in higher dimensions. The more dimensions added, the more challenging it becomes to define the shape of the soap film; however, Uhlenbeck was able to determine the shapes that soap films can take in higher dimensional manifolds. A manifold, simply put, is a curved surface that can be difficult to visualize in higher dimensions. Working with other mathematicians, Uhlenbeck developed methods for using partial differential equations to describe manifold surfaces. Thus, she is now known as one of the pioneers of geometric analysis, a field of mathematics that studies shapes using partial differential equations (Saplakoglu).

 In addition to her research in geometric analysis, Uhlenbeck made a major contribution to gauge theory in mathematical physics by solving the problem of bubble singularities. When studying surfaces and gauge theories, bubble singularities are problematic because it is impossible to do calculus at that point (Saplakoglu). Uhlenbeck found a new coordinate system for four dimensional shapes that removes the singularities, and she provided a proof of her “removable singularities” theorem (Klarreich).

 After reading about Uhlenbeck’s research and accomplishments, it became evident why they say that she revolutionized mathematics as well as inspired many mathematicians, including myself. It was interesting to study minimal surfaces in 3-dimensional space, but I hope that I can explore them in higher dimensions later in my education.

Applications of Minimal Surfaces in Architecture and Nature

Outside of the world of mathematics, applications of minimal surfaces and soap films are seen in architecture and nature. For example, architect Sergio Musmeci’s goal was to build a bridge over the Basento River in Italy that minimized material; thus, he used soap films to help him design the bridge. The Musmeci Bridge, shown below, was constructed in 1972-75 (Adriaenssens, Gabriele, Magrone, & Tomasello).
The soap film that inspired the structure of the Musmeci Bridge.
Image from Revisiting the form finding techniques of Sergio Musmeci

The Musmeci Bridge.
Image from Form Finding Lab

 Likewise, architect Frei Otto used soap films to create the shape of the 1972 Olympic roof in Munich (Emmer).
The roof of the 1972 Olympic Stadium.
Image from Archdaily

 In nature, minimal surfaces appear in “diblock copolymers, smectic liquid crystals, crystallography, semiconductor technology,...even in the cuticular structure in the wing scales of certain insects” (Pérez).
Minimal surfaces are seen in nature.
Image from A New Golden Age of Minimal Surfaces

Although we may not realize it, the structures of buildings around us may have been inspired by minimal surfaces, and they also seem to be hidden in nature. Many people tend to believe that mathematics is not relevant to the "real world," but that is certainly not true about minimal surfaces.

References

Adriaenssens, S., Gabriele, S., Magrone, P., & Tomasello, G. (2016). “Revisiting the form finding techniques of Sergio Musmeci: the bridge over the Basento river: beyond their limits.” In Structures and architecture. Retrieved from https://www.researchgate.net/publication/305872003_Revisiting_the_form_finding_techniques_of_Sergio_Musmeci_The_bridge_over_the_Basento_river_Beyond_their_Limits

Emmer, M. (2013). Minimal surfaces and architecture: new forms. Nexus Network Journal, 15(2), 227-239. https://link.springer.com/content/pdf/10.1007/s00004-013-0147-7.pdf.

Klarreich, E. (2019). Karen Uhlenbeck, uniter of geometry and analysis, wins Abel Prize. Quanta Magazine. Retrieved from https://www.quantamagazine.org/karen-uhlenbeck-uniter-of-geometry-and-analysis-wins-abel-prize-20190319/

Pérez, J. (2017). A new golden age of minimal surfaces. American Mathematical Society, 64(4), 347-358. https://www.ams.org/journals/notices/201704/rnoti-p347.pdf.

Saplakoglu, Y. (2019). Karen Uhlenbeck just won one of math's most prestigious prizes. Here's why her work is so important. Live Science. Retrieved from https://www.livescience.com/65047-karen-uhlenbeck-wins-abel-prize-mathematics.html

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