The geometry of a surface embedded in a 3-manifold with constant mean curvature

While the theory of constant mean curvature surfaces is a purely mathematical one, such surfaces overtly present themselves in nature and are studied in many material sciences. This makes the theory more exciting and adds value to the project. If we take a closed wire and dip it in and out of soapy water, the soap film that forms across the hoop is in fact a minimal (constant mean curvature equal to zero) surface, and the physical properties of soap films were already studied by Plateau in the 1850s. Among all surfaces spanning a given boundary, a soap film is the one with least area. While the air pressure on the sides of a soap film is equal and constant, if we change the pressure on one side, for instance by blowing air on it, the new surface that we obtain is a soap bubble. Mathematically, soap bubbles are constant mean curvature surfaces, and they minimize area under a volume constraint.

This project aims to describe several geometric properties of minimal and constant mean curvature surfaces. Roughly speaking, I intend to prove that constant mean curvature surfaces which satisfy certain weak geometric conditions cannot bend too much. I intend to construct examples to show that the previous results are, in a sense, sharp. Finally, I intend to study the limits of certain sequences of minimal surfaces.

While I am mostly interested in the theoretical aspects of the subject, it is true that constant mean curvature surfaces overtly present themselves in nature and are studied in many material sciences. A key property of constant mean curvature surfaces is that they minimize area under a volume constraint. When there is not a volume constraint, such surface are called minimal surfaces and their constant mean curvature is equal to zero. Among all surfaces spanning a given boundary, a minimal surface is the one with least area. Thanks to their "minimizing properties," questions about the possible shapes of constant mean curvature surfaces are of great interest to engineers, physicists and material scientists, and it is impossible to compile an exhaustive list of applications.

For example, in architecture, constant mean curvature surfaces are of interest for aesthetic reasons, a key example being the Olympiapark in Munich, Germany, but also for practical reasons. Tensile membrane structures are modeled after minimal surfaces while pneumatic structures, that is membrane structures that are stabilized by the pressure of compressed air such as inflatable domes and enclosures, are modeled by constant mean curvature surfaces. Constant mean curvature surfaces are also of interest to engineers for several reasons, one of them being their close relation to foam structures. Foam structures are renowned for their elastic properties and are employed by engineers in a wide range of situations, for instance as insulators. Each soap film of a foam structure is a constant mean curvature surface. Finally, an example of applications of constant mean curvature surfaces in medicine is provided by the fact that they serve as model for the heart wall muscle fibers, explaining how the heart optimizes ventricular ejection while minimizing fiber length.