Geometric Analysis Gallery

Minimal surfaces

The rigorous construction of new surfaces in geometry is comparable to the discovery of new species in biology, the synthesis of new molecules in chemistry or the detection of new particles in physics. Exploring whether surfaces with specific properties exist helps us understand how our ambient space influences which shapes are possible. A desirable property for surfaces is being critical for the area functional, that is being a minimal surface. This means that their area cannot be decreased to first order by any small variation of the surface. Equivalently, the mean curvature of the surface vanishes everywhere, signifying that at every point, the surface has a tangential direction with the largest positive curvature and a second tangential direction, orthogonal to the first, with an equal amount of negative curvature. As a result of this equality, minimal surfaces are particularly aesthetic and possess rich geometric and analytic properties. The search for and analysis of these surfaces has been the focus of the research of some of the most renowned mathematicians of the past three centuries.

The catenoid was the first (nonplanar) minimal surface discovered by Euler. Meusnier's helicoid and Scherk's minimal surfaces were later added to the list of known examples. Enneper, Schwarz, Riemann, and Weierstrass further developed the theory. Plateau demonstrated that minimal surfaces appear in nature as surfaces with the least potential surface energy. He experimentally created them by dipping wires in the form of closed curves into soap water. The famous problem about the existence of a minimal surface with an arbitrarily given Jordan curve as boundary was named after him. Its solution, achieved independently by Douglas and Radó, is a milestone in the history of mathematics. The corresponding Neumann problem, proposed by Courant, asks for free boundary minimal surfaces, i. e. minimal surfaces that meet a given constraint surface orthogonally along their boundary. Such surfaces naturally emerge as interfaces between different fluids of equal pressure and can be used to model cellular membranes. Even in very simple ambient manifolds, many fundamental questions about the existence, uniqueness, and instability of free boundary minimal surfaces remain open.