Sharp Eigenvalue Inequalities and Minimal Surfaces

This project is broadly concerned with the subject of spectral geometry, which studies the relation between certain analytic invariants (called spectra) of objects 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.

More specifically, this project deals with the spectral geometry of special shapes such as soap films. The geometry of soap films is a central subject of modern geometric analysis with applications far beyond the one suggested by their name, most prominently to mathematical theory of general relativity. Spectral geometry of soap films is an exciting novel area of research on the interface of spectral theory and geometry.

The inherent interdisciplinary nature of this field results in a fruitful exchange of ideas between spectral theory and geometric analysis. The goal of the project is to further develop the techniques and ideas introduced the recent years, to investigate their applications beyond spectral geometry and to interest young researchers in this new promising subject. The project also includes summer research opportunities for undergraduate students.

The project is devoted to the study of sharp isoperimetric inequalities for eigenvalues of natural geometric operators on manifolds. The underlying principle of the field is the correspondence between optimal metrics for such inequalities and minimal submanifolds. As a result, a variety of methods can be applied to this problem and advances in the field often result in interesting information on the geometry of minimal submanifolds.

The following four problems are identified as the focus of the project. The first goal is to extend PI’s prior results on the explicit form of the sharp isoperimetric inequality for all Laplacian eigenvalues on the projective plane to the case of more complicated surfaces, starting with a torus. The second project will study the regularity and stability of optimal metrics.

This will be achieved using the energy min-max characterization of the optimal eigenvalue inequalities obtained in collaboration with D. Stern. The third goal is to investigate the asymptotic behavior of optimal metrics for the first Steklov eigenvalue as the number of boundary components becomes unbounded.

Numerical simulations predict that a novel geometric phenomenon manifests itself in this regime: the corresponding free boundary minimal surfaces converge to a closed minimal surface in the boundary. The aim is to verify this theoretically and further understand the mechanisms of such convergence. Finally, the project will develop the theory of isoperimetric inequalities for eigenvalues of Dirac operator following the framework of Laplace and Steklov eigenvalues.

Broader impacts include co-organizing conferences and mentoring undergraduate students in summer research projects.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.