Mathematics Colloquia and Seminars

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Among n-dimensional regions with fixed volume, which one has the least boundary? This question is known as an isoperimetric problem; its nature depends on what is meant by a "region". I will discuss variations of an isoperimetric problem known as the generalized Cartan-Hadamard conjecture: If Ω is a region in a complete, simply connected n-manifold with curvature bounded above by κ≤0, then does it have the least boundary when the curvature equals κ and Ω is round? This conjecture was proven when n = 2 by Weil and Bol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. In joint work with Benoit Kloeckner, we generalize Croke's result to most of 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 a planet to maximize gravity at a single point, such as the place where the Little Prince stands on his own small planet.