The objectives of this project are separated into 3 strongly interconnected areas, spectral geometry, index theory and geometric flows; their unifying theme is the analysis on singular spaces.
The project will treat spectral geometric questions, index theory and geometric flows using the currently available parabolic microlocal methods on simple edge spaces with constant indical roots. This includes:
- Cheeger-Müller Theorem on spaces with even codimension singularities.
- Bergman kernel asymptotics on edges and quantum Hall effect.
- Spectral geometry on edges with variable indicial roots.
- Extension of spectral geometry to stratified spaces.
- Index theory, eta and Cheeger-Gromov rho invariants.
- Long time existence and stability of the singular Ricci flow.
- The porous media equation on edge spaces.
We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M,g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold