Connections Workshop: New Frontiers in Curvature & Special Geometric Structures and Analysis: Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into Euclidean space
Presenter
August 22, 2024
Keywords:
- Riemannian geometry
- curvature
- geometric flow
- special holonomy
- Minimal surface
- general relativity
- K¨ahler geometry
MSC:
- 32-XX - Several complex variables and analytic spaces {For infinite-\newline dimensional holomorphy
- see also 46G20
- 58B12}
- 53-XX - Differential geometry {For differential topology
- see 57Rxx
- for foundational questions of differentiable manifolds
- see 58Axx}
- 58-XX - Global analysis
- analysis on manifolds [See also 32Cxx
- 32Fxx
- 32Wxx
- 46-XX
- 53Cxx] {For nonlinear operators
- see 47Hxx
- for geometric integration theory
- see 49Q15}
- 83-XX - Relativity and gravitational theory
Abstract
We will prove that given an n-dimensional integral current space and a 1-Lipschitz map, from this space onto the n-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani. (Joint work with G. Del Nin).