Introductory Workshop: New Frontiers in Curvature: Harmonic maps and rigidity
Presenter
August 28, 2024
Keywords:
- mean curvature flow
- Ricci flow
- fully nonlinear flows
- general relativity
- mass
- Ricci curvature
- scalar curvature
- sectional curvature
- symmetry
- Riemannian geometry
- group actions
- minimal surfaces
- stability and index
MSC:
- 35J47 - Second-order elliptic systems
- 35K40 - Second-order parabolic systems
- 49J35 - Existence of solutions for minimax problems
- 49Q05 - Minimal surfaces and optimization [See also 53A10
- 58E12]
- 53A10 - Minimal surfaces in differential geometry
- surfaces with prescribed mean curvature [See also 49Q05
- 49Q10
- 53C42]
- 53C20 - Global Riemannian geometry
- including pinching [See also 31C12
- 58B20]
- 53C21 - Methods of global Riemannian geometry
- including PDE methods
- curvature restrictions [See also 58J60]
- 53C22 - Geodesics in global differential geometry [See also 58E10]
- 53C23 - Global geometric and topological methods (à la Gromov)
- differential geometric analysis on metric spaces
- 53C30 - Differential geometry of homogeneous manifolds [See also 14M15
- 14M17
- 32M10
- 57T15]
- 53C43 - Differential geometric aspects of harmonic maps [See also 58E20]
- 53C50 - Global differential geometry of Lorentz manifolds
- manifolds with indefinite metrics
- 53E20 - Ricci flows
- 58C40 - Spectral theory
- eigenvalue problems on manifolds
- 58D19 - Group actions and symmetry properties
- 58E05 - Abstract critical point theory (Morse theory
- Lyusternik-Shnirel'man theory
- etc.) in infinite-dimensional spaces
- 58E15 - Variational problems concerning extremal problems in several variables
- Yang-Mills functionals [See also 81T13]
- etc.
- 58J35 - Heat and other parabolic equation methods for PDEs on manifolds
- 83C05 - Einstein's equations (general structure
- canonical formalism
- Cauchy problems)
- 83C15 - Exact solutions to problems in general relativity and gravitational theory
- 83C20 - Classes of solutions
- algebraically special solutions
- metrics with symmetries for problems in general relativity and gravitational theory
- 83C57 - Black holes
- 83C60 - Spinor and twistor methods in general relativity and gravitational theory
- Newman-Penrose formalism
- 83C75 - Space-time singularities
- cosmic censorship
Abstract
We examine harmonic maps into non-positively curved metric spaces, with a focus on their regularity when mapping into Euclidean buildings. We extend the regularity results of Gromov and Schoen by considering buildings that are not necessarily locally finite. As an application, we prove a superrigidity theorem for algebraic groups, which generalizes the rank 1 p-adic superrigidity results of Gromov and Schoen. This work also provides a geometric framework for Bader and Furman's generalization of Margulis' higher rank superrigidity theorem, bridging the gap between geometric and algebraic approaches to rigidity phenomena.