Isoperimetric study of initial data for the Einstein equations

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.

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. With the support of the FWF START Award, my team and I have been able to make several contributions that deepen our understanding of how surfaces with distinct geometric properties such as cell membranes (least bending energy), soap films (least area), or isoperimetric surfaces (largest enclosed volume at a given amount of boundary area) in initial data sets for the Einstein field equations witness and often constrain physical features of the evolving spacetime. These include the resolutions of several longstanding conjectures in the mathematical theory of general relativity, some of which date back to the first golden age of variational methods in the field in the late seventies and eighties. The success of this FWF START Award has laid the foundations for several newly established, permanent research positions in analysis, geometry, and general relativity at the University of Vienna.