Isoperimetric study of initial data for the Einstein equations
Isoperimetric study of initial data for the Einstein equations
Disciplines
Mathematics (90%); Physics, Astronomy (10%)
Keywords
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Isoperimetric Problem,
Asymptotically Flat Manifolds,
Scalar Curvature,
Minimal Surfaces,
Positive Mass Theorem,
Constant Mean Curvature (Cmc) Surfaces
The question of isoperimetry - How much area can one enclose with a given amount of perimeter? - goes back to antiquity. Indeed, in Virgil`s Aeneid, Queen Dido was challenged to claim as large a piece of land on the northern shore of Africa as she could with string cut from a piece of oxhide. By choosing a circular perimeter for what would become the famed city of Carthage, Dido found the optimal solution and rose to be one of the first great heroines of mathematics. A circumstance that works in Dido`s favor is that the surface of the earth is positively curved, rather than flat. (This is elementary. Let me challenge you to think about why this is to her advantage for a moment.) It turns out that positive curvature is characterized by the property that a small disk - the set of points within a given distance from a point that is - with a given perimeter encloses ever so much more area than a flat disk. In other words, the isoperimetric problem in the small prefers positive curvature. According to the initial value formulation of general relativity, all that is future and all that is past is contained in a glimpse of the spacetime: an initial data set. Under natural conditions, the (scalar) curvature of these initial data geometries is non-negative. There is a deep and powerful relationship between physical properties of spacetime such as its mass, its center of mass, or the shape of its event horizon on the one hand and the isoperimetric problem in the small and in the large for initial data of the spacetime on the other hand. The goal of my research is to investigate this relationship in depth and to use it to shed light on some questions at the intersection of geometry and general relativity.
- Universität Wien - 100%
- Jan Metzger, Universität Potsdam - Germany
- Christopher Nerz, Universität Tübingen - Germany
- Simon Brendle, Columbia University New York - USA
- Otis Chodosh, Stanford University - USA
- Gregory Galloway, University of Miami - USA
Research Output
- 19 Citations
- 6 Publications
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2017
Title Some remarks on energy inequalities for harmonic maps with potential DOI 10.1007/s00013-017-1049-9 Type Journal Article Author Branding V Journal Archiv der Mathematik Pages 151-165 Link Publication -
2017
Title On Conservation Laws for the Supersymmetric Sigma Model DOI 10.1007/s00025-017-0756-7 Type Journal Article Author Branding V Journal Results in Mathematics Pages 2181-2201 Link Publication -
2025
Title Multisoliton solutions for equivariant wave maps on a 2+1 dimensional wormhole DOI 10.1103/physrevd.111.024006 Type Journal Article Author Bizon P Journal Physical Review D Pages 024006 -
2025
Title Instability of nonlinear scalar field on strongly charged asymptotically AdS black hole background DOI 10.1103/physrevd.111.064017 Type Journal Article Author Ficek F Journal Physical Review D Pages 064017 Link Publication -
2022
Title Characteristic approach to the soliton resolution DOI 10.1088/1361-6544/ac7b04 Type Journal Article Author Bizon P Journal Nonlinearity Pages 4585-4598 Link Publication -
2021
Title Initial Data Rigidity Results DOI 10.1007/s00220-021-04033-x Type Journal Article Author Eichmair M Journal Communications in Mathematical Physics Pages 253-268 Link Publication