Recent progress on geometric analysis and Riemannian geometry: Toral symmetries of collapsed ancient homogeneous Ricci flows
Presenter
October 24, 2024
Keywords:
- mean curvature flow
- Ricci flow
- fully nonlinear flows
- general relativity
- mass
- Ricci curvature
- scalar curvature
- sectional curvature
- symmetry
- Riemannian geometry
- groups actions
- minimal surfaces
- stability
MSC:
- 35-XX - Partial differential equations
- 49-XX - Calculus of variations and optimal control
- optimization
- 53-XX - Differential geometry
- 58-XX - Global analysis
- analysis on manifolds
- 83-XX - Relativity and gravitational theory
Abstract
Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle. (Joint work with F. Pediconi and S. Sbiti.)