The Cartan-Hadamard conjecture and the Little Prince

摘要

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 <= kappa <= 0, then partial derivative 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 not equal 0 is a new result. As Croke implicitly did, we relax the curvature condition K <= kappa to a weaker candle condition Candle(kappa) or LCD(kappa).