Stephane Sabourau (U. Paris-Est)

Title: Systolically extremal metrics on nonpositively curved surfaces
Abstract: The regularity of systolically extremal surfaces (i.e., surfaces of minimal area with fixed systole) is a delicate problem already discussed by M. Gromov in the 80's. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don't have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. Joint work with M. Katz.