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.
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.
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