Stable hypersurfaces with prescribed mean curvature

The area of a surface governs many physical phenomena. Nature tends to optimise shapes by finding equilibrium positions dictated by a minimality property- roughly speaking, it prefers to use as little area as possible. Well-known examples of this phenomenon are soap films. As early as the mid 19th century, the physicist Plateau conducted experiments in which he immersed a closed wire in and out of a soap solution. The resulting soap film is a minimal surface, i.e. it locally minimizes area among surfaces spanning the given wire (it avoids wasting soap). Of particular interest are configurations of ``stable'' equilibrium, i.e. under any slight perturbation the film will go back to its initial position. Similarly, in the case of soap bubbles, it is again a minimality property of area that dictates their shape (e.g. spherical bubbles), with the difference that this time the minimality is achieved under the constraint of a fixed enclosed volume (how much air the bubble contains): the surface obtained is characterized by having constant mean curvature (CMC). The mean curvature of a soap film or bubble is a geometric quantity that is proportional to the pressure difference on the sides of the film.
The optimising behaviour observed in these examples is ubiquitous in nature (for example, bees use hexagonal cells because this requires the minimal amount of wax for tiling a planar portion); the following is a further example, taken from capillarity theory, and it is very relevant to the present project.

Consider a stable equilibrium configuration for a liquid that is surrounded by air, subject to surface tension and to the action of external body forces, such as gravitational energy. By a principle of energy optimization, the equilibrium configuration is once again dictated by a partial differential equation whose geometric content is to prescribe the mean curvature of the interface (the surface that separates liquid and air). More precisely, in the absence of gravity or other external forces, the condition is that the mean curvature is constant (CMC surfaces); in the presence of a non-zero potential, for example, a gravitational one, the mean curvature is prescribed up to an additive constant by the value of the potential.

Modern geometry is not limited to surfaces in three-dimensional space and this has allowed, and will for time to come, far-reaching applications, from relativity theory and black holes to engineering. It is therefore natural to introduce hypersurfaces (a generalization to arbitrary dimensions of a surface in three-dimensional space) of dimension n that sit in an ambient space of dimension n+1. In mathematics this ambient space is a Riemannian manifold, i.e. a space with compatible notions of length and angle that permit the computation of area, volume, etc.

In this project I study stable hypersurfaces whose mean curvature is prescribed by a given function on the ambient Riemannian manifold (special cases of which include minimal and constant-mean-curvature hypersurfaces). The project aims to address the fundamental geometric question of existence of closed hypersurfaces of this type in arbitrary closed Riemannian manifolds, employing an analytic framework (regularity and compactness results) that I recently developed. The successful completion of this project will be a pathway towards a more complete understanding of interfaces between liquids and air (as in the capillarity model above).

The project is completely self-contained, it addresses a fundamental geometric problem with an underlying and fundamental analytic component and an expected impact on applied problems. In addition to the PI, it will involve a PDRA that will be chosen and trained, thereby adding expertise in the UK in the area of geometric analysis. Moreover, part of the proposed research will be conducted in collaboration with other UK-based mathematicians. In view of these considerations, the successful completion of the proposal will foster a UK-based ``school'' with expertise in hot topics at the interface between partial differential equations and geometry, in line with the aims outlined in the 2010 International Review of Mathematical Sciences and the 2016 EPSRC Mathematical Sciences Community Overview Documents. The importance of this area of research can be stressed by recalling a few major breakthroughs and awards over the past dozen years, starting with Perelman's proof of the geometrization and Poincare conjectures (Fields medal 2006), until the more recent proofs by Marques-Neves of the 50-year-old Willmore conjecture, by Brendle--Schoen of the 1/4-pinching conjecture, and the very recent proof of the positive mass theorem in all dimensions by Schoen-Yau; the award to Nirenberg and Nash of the Abel Prize in 2016 for ``seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis'' testifies the importance placed on progress in the area.

From an economic and societal perspective, the creation of a "school", the knowledge produced and the techniques developed will be key attractive features for future MSci, MPhil and PhD students interested in geometric analysis to choose the UK over other world-leading institutions based abroad. At the same time, the completion of the present project will provide an occasion to train students in geometric analysis, by introducing ad hoc 4th year and graduate courses motivated by the achievements obtained; this will enhance the competitiveness of home-grown talent in view of future hirings in the UK, a direction encouraged by the 2010 International Review of Mathematical Sciences. Implicitly, by reinforcing the leading role of the UK in geometric analysis, the success of the project will make academic institutions in the UK less prone to losing their leaders to abroad institutions, in agreement with the objectives of the 2016 EPSRC Mathematical Sciences Community Overview Documents.

The exchanges between the analytic and geometric aspects in this project will provide an occasion for the team to interact with leading researchers (based in the UK and abroad) from the analysis and applied communities. The 3-day workshop proposed will provide a platform for such interactions. In addition to the dissemination component, with the presence of internationally affirmed mathematicians, the workshop will feature an outlook component: the activities will provide an occasion to identify future directions of research that are of interest to several mathematical communities and will strengthen cross-disciplinary links. At the same time, the workshop will be beneficial for the London EPSRC Centre for Doctoral Training, both as a learning opportunity for students and to enhance the identification of new and interesting research projects, including those for potential PhD theses.