Stability for nonlocal curvature functionals

In Geometry and Analysis, the curvature of and its connection to the shape of a surface is one of the most widely researched topics. During the past 20 years, their significance has been highlighted through Fields Medal awards to Perelman (2006) and Figalli (2018). This project is about analytical and geometrical aspects of curvature.

Let us first have a look at the geometric aspects of curvature. On the local level, the curvature of a surface at a point can intuitively be visualized by the amount and direction of the bending of a surface near this point. This notion of curvature, in the following called "local curvature", is the classical notion and for more than 200 years it has been a major theme in Analysis, how local curvature determines the shape of the surface. For example, if the local curvature is constant among all of the surface points in a suitable sense, then the surface must be a flat plane or a piece of a round sphere. Relaxing the hypothesis in this statement, it is also true that if the curvature is "almost" constant, then the surface must be "close" to a round sphere, where those terms have to be defined precisely to make a rigorous mathematical statement. Questions of this sort are on the edge of current research.
This project is about extending questions of the described flavour to new notions of curvature. For example, another notion of curvature of a closed surface could be, how big a ball touching at a surface point may at most be in order to fit into the region that is enclosed by the surface. Contrary to the local curvature, which only depends on the shape of the surface "nearby" a points, this new notion of curvature depends on the global shape of the surface and can not be measured by small inhabitants of the surface. Hence we call such notions "non-local curvature". Research in this area has just started and most results have been developed during the past 10 years. There are many possible ways to define notions of non-local curvature and for a few particular examples, this project intends to explore their connection to the global shape of the surface.

Coming to the analytical aspects of curvature, the local curvature is resembled by the second derivative of a parametrisation of the surface, which should not come as a surprise given that it represents the bending of the surface. Hence this branch of research is closely related to the study of partial differential equations of second order.
In Analysis there is a non-local version of derivatives, which are usually call "fractional derivatives". Those are defined using suitable integrals over the whole domain of a function. Hence, as local curvature is defined via classical derivatives of a function, by analogy it is tempting to define non-local curvature by fractional derivatives of a function. This is precisely what we are aiming to explore in this project and we hope it will trigger broad interest in the scientific community working in Geometric Analysis.