Seminar

Among all n-dimensional manifolds D with boundary, with some fixed volume, and with curvature bounded above by some constant κ, which one has the least surface area of its boundary? The nice answer that one might expect is a round ball with constant curvature κ.  In order to attain any positive lower bound on the surface area, we need extra assumptions concerning the topology and geometry of D.

The generalized Cartan-Hadamard conjecture is this isoperimetric inequality if D is a domain in a complete, simply connected manifold M with the same curvature bound κ and κ is non-positive.  This conjecture was proven by Weil and Bol in dimension 2, by Kleiner in dimension 3, and by Croke in dimension 4 when κ=0.

I will discuss a new interpretation of Croke's argument based on optimal transport and linear programming.  This interpretation allows a generalization, in dimensino 4, to positive κ and a partial generalization to negative κ.  This is joint work with Benoit Kloeckner.