The Cartan–Hadamard conjecture and the Little Prince

Abstract

The generalized Cartan–Hadamard conjecture says that if \Omega is a domain with fixed volume in a complete, simply connected Riemannian n -manifold M with sectional curvature K \le \kappa \le 0 , then \partial\Omega has the least possible boundary volume when \Omega is a round n -ball with constant curvature K=\kappa . The case n=2 and \kappa=0 is an old result of Weil. We give a unified proof of this conjecture in dimensions n=2 and n=4 when \kappa=0 , and a special case of the conjecture for \kappa < 0 and a version for \kappa > 0 . Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for n=4 and \kappa=0 . The generalization to n=4 and \kappa \ne 0 is a new result. As Croke implicitly did, we relax the curvature condition K \le \kappa to a weaker candle condition Candle} (\kappa) or LCD} (\kappa) . We also find counterexamples to a naïve version of the Cartan–Hadamard conjecture: For every \epsilon > 0 , there is a Riemannian \Omega \cong B^3 with (1-\epsilon) -pinched negative curvature, and with |\partial\Omega| bounded by a function of \epsilon and |\Omega| arbitrarily large. We begin with a pointwise isoperimetric problem called "the problem of the Little Prince". Its proof becomes part of the more general method.