Analysis of Spectral Invariants on Manifolds

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Proposal 0302647 P.I.: Kate Okikiolu (UCSD) Analysis of Spectral Invariants on Manifolds: Abstract Associated to a Riemannian manifold, there exist natural geometric operators such as the Laplace-Beltami operator, and one can study the relationship between the geometry of the manifold and the spectra of these operators. (A famous question on this theme is ``can you hear the shape of a drum?".) This leads to the study of spectral invariants of geometric operators, which are geometric invariants of the manifold. The zeta invariants of geometric elliptic operators form a family of such spectral invariants which includes in particular the determinant of the Laplacian and certain integrated local invariants such as the Yamabe functional. It is well known that among metrics of a given volume and conformal class, the Yamabe functional is extremized at a metric of constant scalar curvature, and the same is true for the determinant of the Laplacian on surfaces. This raises many questions regarding how the determinant of the Laplacian behaves in higher dimensions, what happens for different types of Laplacian, and how other zeta invariants behave. There has already been progress on several of these questions, the motivation being both to understand the behavior of specific zeta invariants and to investigate possible applications of zeta invariants to geometry. Okikiolu proposes to study a number of issues related to the existence, uniqueness and behavior of critical metrics for the determinant of the Laplacian and other zeta invariants, including questions concerning global upper or lower bounds, gradient flow, behavior across conformal classes, behavior of model problems, and development of the basic analytic theory of zeta invariants. In addition, Okikiolu proposes to work on related questions concerning spectral invariants of Toeplitz operators on manifolds, and in a somewhat different direction, on the problem of extending the Verlinde formulas from quantum field theory. Understanding the relationship between the geometry of a space and the spectra of natural geometric differential operators on the space is a problem which arises in a number of branches of science. Individual eigenvalues are hard to analyze and often the geometry of a space is more clearly reflected by certain weighted averages of the eigenvalues such as zeta invariants. In particular, determinants of Laplacians on manifolds have been studied and applied in several fields of mathematics and physics including topology, quantum field theory, string theory, algebraic geometry, and conformal geometry. The research proposed here should lead to a more complete mathematical theory of zeta invariants for geometrical and physical applications.