Eigenvalue optimisation and Minimal surfaces

ongoing

This project is positioned on the interface between spectral theory and geometric analysis, and is concerned with the relation between certain analytic invariants of objects (called spectra) and the shape (or geometry) of these objects. A classical example of a spectrum is a Laplacian spectrum. It is a sequence of numbers encoding the information about many physical properties of an object such as elasticity, heat and sound propagations and many others. Specifically, this project deals with the geometry of special shapes such as soap films. The geometry of soap films (the so-called minimal surfaces) is a central subject of modern geometric analysis with applications far beyond the one suggested by their name, most prominently to topology and theory of general relativity. More precisely, the project is devoted to the study of two classical topics: sharp inequalities for eigenvalues of geometric operators on manifolds and minimal surfaces. The underlying principle of the field is the correspondence between minimal surfaces and optimal metrics for such inequalities. This correspondence is both a source of many fascinating intradisciplinary connections (e.g. with analysis, geometry, algebra and theory of integrable systems) and a powerful methodological tool. A variety of methods from across mathematics can be applied to this problem and recent advances have resulted in a resolution of several classical problems on either side of the correspondence. The present project aims to build on this success story by targeting long-standing conjectures in the following directions: 1) Using the correspondence to construct many novel examples of minimal surfaces, study their geometry, and look for new patterns and phenomena; 2) Applying methods of geometric analysis to find sharp eigenvalue inequalities for classical eigenvalues problems; 3) Extending the realm of potential applications by exploring different eigenvalue problems for similar connections to geometric analysis.