Minimal and constant mean curvature surfaces: their geometric and topological properties.

While the theory of minimal and constant mean curvature (CMC) 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. If we take a closed wire and dip it in and out of soapy water, the soap film that forms across the loop is in fact a minimal surface and the physical properties of soap films were already studied by Plateau in the 1850s. The air pressure on the sides of soap films is equal and constant. However, if we change the pressure on one side, for instance by blowing air on it, the new surface that we obtain is what we call a soap bubble. A soap bubble is a CMC surface. More precisely, minimal and CMC surfaces are, respectively, mathematical idealisation of soap films and soap bubbles. The mean curvature of a soap film and bubble is a quantity that is proportional to the pressure difference on the sides of the film. The value of the pressure difference, and therefore of the mean curvature, is zero for a soap film/minimal surface and it is non-zero constant for a soap bubble/CMC surface. Since the pressure inside a small bubble is greater than the pressure inside a big one, the constant mean curvature of a small bubble is greater than the constant mean curvature of a big one.

Minimal and CMC surfaces also enjoy crucial minimising properties relative to area. Among all surfaces spanning a given boundary, a soap film/minimal surface is one with locally least area; soap bubbles/CMC surfaces locally minimise area under a volume constraint. This project aims to investigate several key geometric properties of minimal and CMC surfaces. Roughly speaking, I intend to prove several results about CMC surfaces embedded in a flat three-dimensional manifold, including area estimates when the surfaces are compact with bounded genus and the ambient manifold is compact. I also plan to study the limits of a sequence of minimal or CMC surfaces embedded in a general three-dimensional manifold.

While my research proposal focuses on fundamental questions in Pure Mathematics, more specifically in the theory of minimal and constant mean curvature (CMC) surfaces, the close connection between minimal and CMC surfaces and nature gives the subject a very concrete perspective and enhances the potential for broader impact. Minimal and CMC surfaces are, respectively, mathematical idealisation of soap films and soap bubbles. Among all surfaces spanning a given boundary, a soap film is the one with locally least area; soap bubbles/CMC surfaces locally minimise area under a volume constraint. Thanks mainly to such "minimising properties,'' questions about the possible shapes of minimal and CMC surfaces are also of great interest to architects, engineers, physicists and material scientists. For instance, the shape of a string of DNA closely resembles that of a double spiralling staircase, that is a minimal surface; it may be that a double spiralling staircase is nature's safe way of storing lots of information.

As part of my project, I intend to study the geometry of certain CMC surfaces that appear in nature. For example, approximations to these special surfaces manifest themselves in studies of Fermi surfaces (equipotential surfaces) in condensed matter physics, and the related constant potential surfaces in crystals. They are also found in material sciences where certain CMC surfaces are seen to closely approximate surface interfaces in certain inhomogeneous mixtures, and as the boundary of the microscopic calcium deposit patterns in sea urchin shells. The geometry of such surfaces arising in nature has profound consequences for the physical properties of the materials in which they occur, which is the reason why they are studied in great detail by physical scientists and why my proposed project might interest a rather broad pool of scientists. While my research proposal focuses on fundamental questions in Pure Mathematics and it is hard for me to formulate a detailed prediction, for the aforementioned reasons a timely completion of the research project could lead to some crucial long-term impact.

The UK is internationally recognised as a leader in Geometry, which is a fundamental part of modern mathematics and has applications throughout the physical and life sciences, engineering and ICT. This said, the EPSRC strategic plan clearly emphasises the need "to strengthen and develop the UK's connections between geometric analysis and partial differential equations.'' This was partially motivated by the IRMS 2010 landscape document for Geometry and Topology indicating such need and recommending that opportunities should be exploited for fruitful collaborations between analysis and other fields of the mathematical sciences, most notably in geometry and topology. This was later reiterated by the EPSRC Pure Maths Workshop (2012). The proposed project studies minimal and CMC surfaces and the theory of such surface is well-renowned for its interaction with the theory of PDEs. In fact, when a CMC surface is given as the graph of a function (always true locally) such graph satisfies a prototypical quasi-linear elliptic PDE. Thus my project provides an opportunity for the UK for strengthening its expertise in this field of research.

The impact of my research project will also come from the training of a PDRA, my research visits and the visits to the UK of my collaborators. In particular, its impact will be enhanced by my and their involvement with the EPSRC funded new CDT at King's College London (jointly with Imperial College and University College London). Students who go on to careers in industry or applied scientific research, including possibly my current PhD student, will greatly benefit from being exposed to our research. Finally, the impact will be facilitated by a series of workshops and a public engagement event that I plan to organise at King's College London.