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EPSRC Reference: EP/S005641/1
Title: Stable hypersurfaces with prescribed mean curvature
Principal Investigator: Bellettini, Professor C
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Researcher Co-Investigators:
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Department: Mathematics
Organisation: UCL
Scheme: New Investigator Award
Starts: 01 September 2018 Ends: 31 August 2023 Value (£): 287,098
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Panel History:
Panel DatePanel NameOutcome
06 Jun 2018 EPSRC Mathematical Sciences Prioritisation Panel June 2018 Announced
Summary on Grant Application Form
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).
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