Introductory Workshop: New Frontiers in Curvature: Spaces with Ricci curvature lower bounds, pt. 1
Presenter
August 26, 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
Ricci curvature lies between the sectional curvature and scalar curvature. It occurs naturally in Einstein equation, Ricci flow, optimal transport. We will review the basic tools (Bochner's formula, comparison inequalities) and survey many geometric inequalities (Neumann eigenvalue comparison, Brunn-Minkowski, Sobolev inequalities etc.), topological properties and examples in the smooth and nonsmooth settings.