SubsidieMeestersComparison and rigidity for scalar curvature
This project aims to develop new tools for studying scalar curvature by unifying existing methods and addressing gaps in techniques related to geometric analysis and differential geometry.
Projectdetails
Introduction
Questions involving the scalar curvature bridge many areas inside mathematics including geometric analysis, differential geometry, and algebraic topology, and they are naturally related to the mathematical description of general relativity.
Methods to Probe Scalar Curvature
There are two main flavours of methods to probe the geometry of scalar curvature:
- One goes back to Lichnerowicz and uses various versions of index theory for the Dirac equation on spinors.
- The other is broadly based on minimal hypersurfaces and was initiated by Schoen and Yau.
On both types of methods, there has been tremendous progress over recent years sparked by novel quantitative comparison and rigidity questions due to Gromov and by ongoing attempts to arrive at a deeper geometric understanding of lower scalar curvature bounds.
Unified Standpoint
In this proposal, we view established landmark results, such as the non-existence of positive scalar curvature on the torus, together with the more recent quantitative problems from a conceptually unified standpoint. Here, a comparison principle for scalar and mean curvature along maps between Riemannian manifolds plays the central role.
Development of New Tools
Guided by this point of view, we aim to develop fundamentally new tools to study scalar curvature that bridge long-standing gaps in between the existing techniques. This includes:
- A far-reaching generalization of the Dirac operator approach expanding upon techniques pioneered by the PI.
- Novel applications of Bochner-type methods.
Study of Comparison Problems
We will also study analogous comparison problems on domains with singular boundaries motivated by:
- A first synthetic characterization of lower scalar curvature bounds in terms of polyhedral domains.
- The general quest for extending the study of scalar curvature beyond smooth manifolds.
Rigidity Questions
At the same time, we will treat subtle almost rigidity questions corresponding to the rigidity aspect of our comparison principle.
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.421.111 |
| Totale projectbegroting | € 1.421.111 |
Tijdlijn
| Startdatum | 1-1-2024 |
| Einddatum | 31-12-2028 |
| Subsidiejaar | 2024 |
Partners & Locaties
Projectpartners
- UNIVERSITAET MUENSTERpenvoerder
Land(en)
Vergelijkbare projecten binnen European Research Council
| Project | Regeling | Bedrag | Jaar | Actie |
|---|---|---|---|---|
Geometry and analysis for (G,X)-structures and their deformation spacesThis project aims to advance geometric structures on manifolds through innovative techniques, addressing key conjectures and enhancing applications in topology and representation theory. | ERC Consolid... | € 1.676.870 | 2024 | Details |
Minimal submanifolds in Arbitrary Geometries as Nodal sEts: Towards hIgher CodimensionThis research aims to explore the calculus of variations in higher codimension, focusing on critical points and gradient flows of minimal submanifolds to uncover links between geometry and topology. | ERC Starting... | € 1.420.400 | 2025 | Details |
Geometric Analysis and Surface GroupsThis project aims to explore the connections between curves in flag manifolds and moduli spaces of Anosov representations, focusing on energy functions, volumes, and topological invariants. | ERC Advanced... | € 2.325.043 | 2024 | Details |
Mean curvature flow: singularity formation beyond 2 convexity and applicationsThe project aims to advance the understanding of singularities in mean curvature flow and apply it to general relativity, focusing on bubble-sheet singularities and cosmic no hair conjecture. | ERC Starting... | € 1.499.138 | 2023 | Details |
Anisotropic geometric variational problems: existence, regularity and uniquenessThis project aims to develop tools for analyzing anisotropic geometric variational problems, focusing on existence, regularity, and uniqueness of anisotropic minimal surfaces in Riemannian manifolds. | ERC Starting... | € 1.492.700 | 2023 | Details |
Geometry and analysis for (G,X)-structures and their deformation spaces
This project aims to advance geometric structures on manifolds through innovative techniques, addressing key conjectures and enhancing applications in topology and representation theory.
Minimal submanifolds in Arbitrary Geometries as Nodal sEts: Towards hIgher Codimension
This research aims to explore the calculus of variations in higher codimension, focusing on critical points and gradient flows of minimal submanifolds to uncover links between geometry and topology.
Geometric Analysis and Surface Groups
This project aims to explore the connections between curves in flag manifolds and moduli spaces of Anosov representations, focusing on energy functions, volumes, and topological invariants.
Mean curvature flow: singularity formation beyond 2 convexity and applications
The project aims to advance the understanding of singularities in mean curvature flow and apply it to general relativity, focusing on bubble-sheet singularities and cosmic no hair conjecture.
Anisotropic geometric variational problems: existence, regularity and uniqueness
This project aims to develop tools for analyzing anisotropic geometric variational problems, focusing on existence, regularity, and uniqueness of anisotropic minimal surfaces in Riemannian manifolds.