###Minimal surfaces and eigenvalues of the Laplacian
Project ID: 2228bd1155 (You will need this ID for your application)
Research Theme: Mathematical Sciences
UCL Lead department: Mathematics
Lead Supervisor: Mikhail Karpukhin
Project Summary:
The problem of maximising Laplacian eigenvalues among surfaces of fixed area and fixed topological type has motivated many interesting developments in recent decades, in particular, due to the intimate connection to the classical object of geometric analysis – minimal surfaces. The latter are a fundamental notion in mathematics. Not only do minimal surfaces have fundamental applications in geometry and topology, but they are also ubiquitous in physics e.g., serving as models for event horizons of black holes or describing the formation of soap films. The connection between geometric eigenvalue optimisation and minimal surfaces allows for a fruitful exchange of ideas and methods between spectral theory and geometric analysis. On one hand, one can apply the theory of minimal surfaces and all its methods to the study of sharp eigenvalue inequalities. On the other hand, this connection sheds a different light on minimal surfaces, which leads to a variety of new questions and surprising insights into their geometry.
At the present time, such a correspondence (with all the geometric consequences thereof) is known only for Laplace and Steklov eigenvalue problems. However, there are strong indication that other operators, such as Dirac operator or magnetic Laplacian, possess similar properties. Finding such examples would lead to further geometric applications and would improve our understanding of this phenomenon as a whole. As a result, the project has two main objectives that can be broadly summarised as follows. First, continue studying the correspondence in the case of Laplace and Steklov eigenvalues. Second, investigate other natural eigenvalue problems in search of the connection with classical notions in geometry.
The project is suitable for a student with a strong background in analysis and Riemannian geometry and will be conducted under the supervision of Dr. M. Karpukhin.