Problems in geometric analysis

Dartmouth College, Hanover NH

Investigators

Abstract

Proposal 0306752 PIs: Carolyn Gordon, Scott Pauls, David Webb Title: PROBLEMS IN GEOMETRIC ANALYISIS The focus of the project will be inverse spectral geometry and analysis of Riemannian and sub-Riemannian geometries on nilmanifolds. Inverse spectral geometry is the study of the extent to which the geometry of a Riemannian manifold can be recovered from spectral data. Gordon and Webb will consider constructions of compact Riemannian manifolds with the same Laplace eigenvalue spectrum and compare their local and global geometry. They, along with their collaborators, will also consider the spectrum of Schroedinger operators on line bundles over tori and spectral data for orbifolds. For noncompact manifolds, the relevant spectral data are the scattering resonances and scattering phase. Gordon, Webb, and Pauls, along with Peter Perry, will investigate possible constructions of Riemannian metrics with the same scattering data. They will also consider isoscattering potentials for the Schroedinger operator and isoscattering obstacles. In the area of sub-Riemannian geometry, Pauls will continue working towards a better understanding of variational problems in Carnot groups, focusing on the regularity of minimal surfaces, the calculations of the best isoperimetric constant for the Heisenberg group, extensions of previous work to more general Carnot groups and on problems related to the spectral theory of the subLaplacian (joint with Doyle, Gordon and Webb). He will also continue working with Mike Wolf (Rice University) on two fundamental problems in the theory of harmonic maps. The investigators will address inverse spectral problems, inverse scattering problems, and sub-Riemannian geometry. Inverse spectral geometry is rooted in spectroscopy, the problem of understanding the nature of a system from the characteristic frequencies of light or sound emitted. The investigators will consider various constructions of objects (Riemannian manifolds such as planar domains, balls, or spheres) which have the same spectra and will compare their geometry in order to identify specific geometric properties that are not spectrally determined. In the quantum mechanical description of a particle in a potential, one distinguishes between ordinary bound states and scattering states whose wave functions are nonnormalizable and whose energies can assume a continuum of possible values. Inverse scattering theory seeks to understand as much as possible about the nature of a potential from the scattering behavior exhibited by particles interacting with the potential. The investigators will address this problem by constructing and studying potentials with the same scattering resonances. The investigations in sub-Riemannian geometry are motivated by a wealth of physical phenomena including problems in wheeled robotic control, satellite navigation and stabilization and thermodynamics. At this point in time, relatively little is known about the general theory guiding these types of control phenomena. Because of this, the investigators will focus on gaining a better understanding of the solutions to cost minimization problems on basic model spaces. Specifically, the investigators are focused on constructing area minimizing surfaces in these settings in order to expose some of the fundamental geometric principles governing the solutions to these types of problems.