Cell membranes, like many structures found in nature, naturally tend toward a state of minimal energy. “Determined by the governing laws of physics, this bending energy depends solely on the membrane’s geometric shape and can be expressed by a surprisingly simple mathematical formula,” explains Dr. Christian Scharrer, a postdoctoral researcher at the Institute of Applied Mathematics at the University of Bonn. From a physical point of view, cell membranes are therefore modeled as energy-minimizing surfaces. Microscopic images of red blood cells closely resemble these mathematically predicted energy-minimizing shapes, demonstrating that the explicit mathematical expression for the energy indeed explains their characteristic biconcave shape.
“Things become particularly exciting when surfaces grow highly complex and have many ‘holes’ in them,” Scharrer says. Mathematically speaking, this is referred to as a high genus: a donut, for instance, has genus one, while a pretzel has genus three. “In our Emmy Noether group, we are now investigating what shapes emerge when the number of such holes in energetically optimal surfaces continues to increase. The conjecture is that, in the limit, a so-called minimal surface emerges,” continues the mathematician. Minimal surfaces are shapes that minimize surface area for a given boundary—like soap films, which do so to reduce surface tension as much as possible. If the number of holes increases while the total surface area remains unchanged, the holes must inevitably become smaller and more tightly packed in certain regions. By zooming in further and further until the smallest hole once again appears to be as large as a donut, mathematicians speak of a “blow-up.” This blow-up is expected to be a minimal surface.
High-genus energy minimization represents an important unsolved task in the field of geometric anal-ysis. Through his research, Dr. Scharrer also aims to contribute to one of the major open problems: the classification of minimal surfaces. “At the same time, we want to develop new foundational tools that, in the long run, will prove valuable in other areas of mathematics as well.”
Dr. Christian Scharrer studied mathematics and physics at the University of Potsdam and earned his doctorate in mathematics from the University of Warwick. He then worked as a Fellow at the Max Planck Institute for Mathematics in Bonn before joining the research group of Professor Stefan Müller at the Institute for Applied Mathematics at the University of Bonn. Dr. Scharrer is a member of the Hausdorff School for Mathematics (HSM) and an associate member of the Hausdorff Center for Mathematics (HCM) Cluster of Excellence. He will be leading an Emmy Noether Group in Bonn starting in April 2026. The German Research Foundation has approved up to €850,000 in funding for the research group for an initial term of three years with a one-time three-year extension option, subject to interim evaluation.