# Details of Grant

EPSRC Reference: EP/M024512/1
Title: Minimal and constant mean curvature surfaces: their geometric and topological properties.
Principal Investigator: Tinaglia, Professor G
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: Kings College London
Scheme: Standard Research
Starts: 01 July 2015 Ends: 30 June 2018 Value (£): 244,300
EPSRC Research Topic Classifications:
 Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
 No relevance to Underpinning Sectors
Related Grants:
Panel History:
 Panel Date Panel Name Outcome 03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
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.
Key Findings
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk