The Cartan-Hadamard Problem and the Little Prince

Geometry Topology Seminar
Tuesday, January 20, 2015 - 14:05
1 hour (actually 50 minutes)
Skiles 006
Among n-dimensional regions with fixed volume, which one hasthe least boundary?   This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region".   I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture:  If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round?  This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke.  Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0.   It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.