Project description
Mean Curvature Flow: Unveiling singularities beyond 2-convexity
Geometric flows, notably the Ricci Flow (RF) and Mean Curvature Flow (MCF), serve as powerful tools for complex topology, geometry, and physics problems. While RF has been quite successful in proving the geometrisation conjecture, MCF demonstrates impressive, yet unfulfilled potential in applications involving sub-manifolds in ambient spaces. The EU-funded MCFBeyondAndApp project aims not only to deepen knowledge of how singularities form in MCF, particularly in structures of a single set, but also to explore their use in general relativity. Specifically, it will build on previous works on 'bubble-sheet' singularities in four-dimensional space, targeting a mean convex neighbourhood theorem. Finally, it will employ MCF in Lorentzian spacetime to explore the cosmic no hair conjecture and its relation to de Sitter space.
Objective
Geometric flows as a mean to attack problems in topology, geometry and physics, had been demonstrated to be an extremely powerful tool. The most successful such flow to date is the Ricci flow (RF), which was used in the proof of the geometrization conjecture. The mean curvature flow (MCF) - the most natural geometric flow for sub-manifolds in an ambient space, had also been successfully applied to address such problems. Nevertheless, the most striking potential applications of MCF are still out of reach. The goal of the proposed research is to advance the understanding of the formation of singularities in MCF, and to study a particular application of MCF for general relativity. More concretely, we propose to continue the systematic study of the formation of bubble-sheet singularities in 4-space, initiated by Choi, Haslhofer and the PI, with the goal of obtaining a mean convex neighbourhood theorem in this setting. We also propose to study the formation of singularities more generally, and in particular, the structure of the singular set. The second objective of the proposed research is to employ MCF in Lorentzian spacetime satisfying the Einstein equation with positive cosmological constant to obtain versions of the cosmic no hair conjecture, namely, geometric convergence results to de Sitter space.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques.
- natural sciencesmathematicspure mathematicstopology
- natural sciencesmathematicspure mathematicsgeometry
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Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
91904 Jerusalem
Israel