Riemannian Geometry and Spectral Analysis

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0072154 John W. Lott Riemannian geometry is the study of curved spaces. Examples of such spaces are curves and surfaces in three-dimensional flat space. Riemann showed how to make precise the notion of curvature for a space of arbitrary dimension. Spectral analysis can be roughly characterized as the study of how a space vibrates. More precisely, to a curved space is associated a certain partial differential operator, the Laplacian. Spectral analysis is the study of how the eigenvalues of the Laplacian depend on the underlying geometry of the space. In previous work, the principal investigator obtained relationships between the spectrum of the differential form Laplacian and the geometry of the underlying space, the latter being constrained by upper bounds on its diameter and upper and lower bounds on its curvature. In particular, he characterized when there are uniform upper bounds on the j-th eigenvalue of the p-form Laplacian, and when there are small positive eigenvalues of the p-form Laplacian. He proposes to extend this work in several directions. One direction is to just assume that there is a lower bound on the curvature of the space. New issues arise in this case, as under the assumed geometric constraints, the space can ``collapse'' to a highly singular space of lower dimension. The principal investigator's previous work, in the case of upper and lower curvature bounds, also dealt with the singular spaces that arise in a collapsing limit. However, with just a lower curvature bound, the singular spaces that arise are of a different nature. He also proposes to extend the previous work in the direction of analyzing the spectrum of the Dirac operator, under the geometric assumptions of an upper bound on the diameter of the space and upper and lower bounds on its curvature.