Optimal inequalities in geometry and relativity

Surfaces can be shaped in all sorts of ways. Some shapes are special. Take, for instance, a sphere, which is shaped the same way at every point, or a catenoid, a certain surface of revolution that does not curve on average. The sphere is special in another, more global sense: Among all surfaces enclosing the same volume, it has the least amount of area. It also has the smallest possible amount of bending energy among all closed surfaces. These two global characterizations are in the form of so-called optimal inequalities. These inequalities relate two or more geometric quantities associated with a surface such as area, volume, or bending energy in an optimal way. They are powerful tools in geometry and allow for a seemingly simple description of complicated geometric objects and their properties. The goal of this project is to prove new such inequalities that fundamentally deepen our understanding of geometry. Is the sphere the closed surface whose area increases the least when inflated evenly? Can the catenoid be characterized by an optimal relation between the size of its neck and its slope at infinity? Surprisingly, such questions are relevant beyond the field of geometry and help us uncover subtle physical features of spacetimes obeying the laws of general relativity.