Introductory Workshop: New Frontiers in Curvature: Complexities of the Second Variation of Minimal Surfaces of codimension greater than one, pt. 2
Presenter
August 27, 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
The nontriviality of the normal bundle of a minimal submanifold of codimension greater than one makes the second variation difficult to study because it is not clear how to choose variations that reduce the area of the submanifold. For a 2-D minimal surface, a partial averaging technique is encoded by a complex-valued version of the second variation formula and this has the benefit of bringing complex analytic techniques into play. I will survey the achievements of this technique and discuss open problems in this area. Other results, including ones for minimal hypersurfaces, that place this work in a broader context will also be mentioned.