Geometric Measure Theory
The central object of study of the research group are two-dimensional or higher-dimensional surfaces in flat or curved spaces of three or more dimensions respectively.
The Max Planck research group "Geometric Measure Theory" is based on a cooperation of the Max Planck Society with the University of Potsdam. Within the Albert Einstein Institute it is in close contact with the division “Geometric Analysis and Gravitation”. At the University of Potsdam it cooperates in particular with the research group “Partial Differential Equations”.
The central object of study of the research group are two-dimensional or higher-dimensional surfaces in flat or curved spaces of three or more dimensions respectively. To ease visualisation one may initially think of two-dimensional surfaces in flat three-dimensional space. Classically, such surfaces are mathematically modelled as smooth submanifolds. In the two-dimensional case, they are objects whose shape can be thought to locally look like a suitably deformed and stretched rubber blanket. However, globally, they may exhibit a more complicated structure, as for instance a torus.
In attempting to solve the higher-dimensional Plateau problem the class of smooth submanifolds proves not to be well suited from a mathematical point of view. (In the classical version of the Plateau problem one would like to obtain a surface minimising area amongst all surfaces spanning a given boundary curve.) In this case, it is a well-established procedure to proceed in two steps. Firstly, one constructs an extended class in which the problem may be solved (existence theory). Subsequently, it is proven that the surface which has been obtained in this way in fact posses, at least partly, the smoothness properties characterising the original class (regularity theory).
Concerning the construction of suitably enlarged classes, consisting for example of certain currents or varifolds, a largely satisfactory state has been reached with the help of Geometric Measure Theory in the last hundred years. Moreover, several decisive contributions to regularity theory have been obtained. In this process, the use of Geometric Measure Theory has proven to be indispensable for surfaces of at least three dimensions.
Nonetheless, there still exist some very important open problems in particular in regularity theory. This especially concerns cases in which it is already known that the solution is much less regular than smooth submanifolds. The main objective of the research group is to contribute to the solution of these questions. Here, the task decomposes into two parts. Firstly, powerful, suitably adapted notions of regularity need to be studied and, occasionally, even invented. Subsequently, it has to be proven that solutions to important variational problems in fact possess these properties.
Finally, it should be mentioned that Geometric Measure Theory has found diverse applications in mathematical theory. In Geometric Analysis, apart of the afore-mentioned geometric variational problems, geometric flows should be mentioned, both of which are research topics of the division "Geometric Analysis and Gravitation". Further applications occur for instance in the theoretical treatment of processes such as image reconstruction or crystal growth which appear to be quite different at first sight.