Giovanni Alberti: Frobenius theorem for non-regular sets and currents

Описание: Abstract: Let V be a non-involutive distribution of k-planes in the Euclidean space. Then Frobenius theorem states that there exist no k-dimensional surface S tangent to V.
In this talk I consider some generalization of this statement to weaker notions of surfaces, such as rectifiable sets and currents.
To begin with, I consider a contact subset E of a k-dimensional surface S (that is, S is tangent to V at every point of E), and I ask if E must have null k-dimensional measure. It turns out that the answer depends on a combination of the regularity of S and of the boundary of E: if S is of class C^(1,α) with α larger than a certain critical exponent the answer is positive for every E; on the other hand, if S is only of class C^1 then the answer is positive if E has nite perimeter, and examples show that this requirement is, in some sense, sharp.
More generally, Frobenius theorem holds when S is an integral current. What then if S is a normal current but is not rectifiable? In this case the key is a certain geometric property of the boundary of S. These questions are strictly related to the problem of decomposing a normal current in terms of integral/rectifiable ones.
These results are part of an ongoing research project with Annalisa Massaccesi (University of Padova), Andrea Merlo (University of Fribourg) and Evgeni Stepanov (Steklov Institute, Saint Petersburg).