Introductory Workshop: New Frontiers in Curvature: Minimal surfaces II
Presenter
August 30, 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
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
I will review the basic principles regarding the basic techniques to prove existence of closed minimal hypesurfaces. I will talk about some current state of the art and finish by pointing out some possible new direction.