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Minimal surfaces and biophysics

The study of minimal surfaces whose graph minimises the area among all surfaces within a given contour started long ago. Still, significant advances continue to be made and striking applications encountered in other fields.
Minimal surfaces and biophysics
Minimal surfaces come in two different types: stable and unstable. A minimal surface is called stable if its area is the smallest one among similar surfaces with the same contour. In the case of an unstable minimal surface, even small variations can decrease its area.

Mathematicians working on the EU-funded RSC AND RMCF (Rigidity of scalar curvature and regularity for mean curvature flow) project focused on the second type. Unstable minimal surfaces seemed interesting, but more importantly very little is known about their properties to date.

RSC AND RMCF work was divided into two parts. First, the mathematicians proved the existence of simple unstable minimal surfaces on three-spheres – a higher-dimensional analogue of a sphere – that are positively curved. Then, they found that the area of such minimal surfaces is bounded in an optimal way.

The methodology developed to prove that, besides glomes, any three-manifold has an infinite number of unstable minimal surfaces was used to find which toroidal shape is the least bent. This simple question was posed in 1965 by Thomas Willmore and remained answered.

The Clifford torus seemed like the most promising candidate. Mathematicians found that this special kind of torus sitting inside the unit three-sphere in the Euclidean space of four dimensions covered less area than any given five-parameter family of surfaces.

The RSC AND RMCF findings suggest that the study of unstable minimal surfaces is a promising new direction to explore in geometrical analysis. Furthermore, finding shapes in equilibrium has implications in biophysics research aiming to explain the shape that human blood cells assume.

Not surprisingly, biophysicists have arrived independently at the Willmore conjecture that the Clifford torus is the least bent toroidal shape by observing toroidal vesicles. It is expected that open questions in geometry also appear in different contexts of other branches of science.

Related information

Keywords

Minimal surfaces, biophysics, RSC AND RMCF, curvature, three-spheres, three-manifolds
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