EGIDI, MICHELA (2015) GEOMETRY, DYNAMICS AND SPECTRAL ANALYSIS ON MANIFOLDS
The Pestov Identity on Frame Bundles and Eigenvalue Asymptotics on Graph-like Manifolds. Doctoral thesis, Durham University.
|PDF - Accepted Version|
This dissertation is made up of two independent parts. In Part I we consider the Pestov Identity, an identity stated for smooth functions on the tangent bundle of a manifold and linking the Riemannian curvature tensor to the generators of the geodesic flow, and we lift it to the bundle of k-tuples of tangent vectors over a compact manifold M of dimension n. We also derive an integrated version over the bundle of orthonormal k-frames of M as well as a restriction to smooth functions on such a bundle. Finally, we present a dynamical application for the parallel transport of the Grassmannian of oriented k-planes of M. In Part II we consider a family of compact and connected n-dimensional manifolds, called graph-like manifold, shrinking to a metric graph in the appropriate limit. We describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian acting on differential forms on those manifolds in the appropriate limit. As an application, we produce manifolds and families of manifolds with arbitrarily large spectral gaps in the spectrum of the Hodge Laplacian.
|Item Type:||Thesis (Doctoral)|
|Award:||Doctor of Philosophy|
|Keywords:||frame bundles, frame flows, principal bundle, pestov identity, grassmannian, parallel transport, curvature operator, Hodge-Laplacian, metric graph Laplacian, eigenvalues, convergence, differential forms, graph-like manifolds, metric graphs, asymptotic behaviour, spectrum.|
|Faculty and Department:||Faculty of Science > Mathematical Sciences, Department of|
|Copyright:||Copyright of this thesis is held by the author|
|Deposited On:||18 Nov 2015 09:44|